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FOREWORD

For four days beginning on 2 June 1952, representatives of the a i r o r a f t induatry and the mil i tary aervices met in Washington, Do C. to p h s e n t and hear papers on the deal@ and u t i l i z a t i o n of piloted a i r c r a f t f l i g h t control s p t m r , The eight papara reproduoed i n t h i a report were presented a t tha t meeting,

The lfavy'a appreoiation l a extended to the authora of theae paperr and t o the varioua organization8 they repreaent,

Comment8 or queationa concerning thia report ahould be forwarded to the Chief, Bureau of Aeronaut 108, Attention AE, Washington 25, Do C

' ' Rear Admiral, U8rS Aarirtrurt Chief f o r

Research and Development

CONTENTS

1, Centering Charauter is t ics of Power Boosted Ai rc ra f t Control Systems' (Martin), . , .

2. Helicopter Powered Control System Problem8 (Slkoraky) . . , . . . . , .

3. Fl ight Research on Force Wheel Control (WAN)

4, An Analyeis of Fully-Powered Ai rc ra f t . Control Syatem Inuluding Some Reference t o F l u t t e r Theory (19orthrop) , ,

5 . Parametera f o r the Design of High-Speed Hydraulic Servomotors (NACA) .- , . , .

3.. Improvement of Power Surfaoe Control Systems by S t ruc tu ra l Deflection Compensation (Boeing). . . . , . , ,

7. A Survey of Suggested Mathematical Methods foy the Study of Human P l l o t f s Response (Franklin) . . . . . .

8, Synthesis of F l igh t Control Systems (Purdue).

CENTERING 'CHARACTERISTICS OF POWER BOOSTED AIRCRAFT CONTROL SYSTEMS'

G. H. G O O K E I. E. KASS

The Glenn L. Martin G o . Baltimore, Maryland

CENTERING CHARACTERISTICS OF POWER BOOSTED AIRCRAFT CONTROL SYSTEMS

W i t h i n fi basically different aircraft, the Glenn L. H a r t i n C a g p ~ y h e designed and put into service 30 different pouered surface control system of M c h 1S wore power operated. The remnining fi power boosted surface control eystsaPs have presented an umwual combination of requiremanta for sensitivity, stability, and oenterlng, To meet such requiranants while m&Lntabing a simple, reliable mchanlsm, devoid of functionally oanplicated acoeesaries, the designer muat provide the optimtun system arrangement Md go to a high i degree of reiinemant in designing the elements.

The design considerations i n a powa: boost& Waf t controls ayutcrn are as follows:

1. To provide optimum sensitivity and tbe correct amount of ponm a t all coaPbinatlo~ of speed and load.

2. To prevent any fom of imtab i l i ty such aa ohattor or hunt Yithkl . the meohanism i n spite of the rensitivity and 1w break-atmy iriotion.

3. To obkin a high degme of system raversibility ro that th. mechanism will accurately centor or return fo the t M podtion, ud so that a good forcre gradlent will k f e l t by the pilot a t vexy low aerodynsrPic moments.

Theee considerations are close2 y interrelated by m ~ l l ~ r 6y8tem perpplbtms which generally am determined by one consideration and haw a direct effeot on the other tuo. Consideration #1 determines eyetan rate gain, d rcu i t powor, and valve chprrcteriaticr, each of which has a d i n a t e f f m t upon r tabi l i ty (consideration #2). Stabil i ty ale0 concams the i r lc t ion o r damp- In# throughout the system, the point of application of the actuator, md the method of obtaining proportional feel, each of which vitally af fwte tho csntsrlng charactaristics (considaration #3).

There a n definite perfonaance requirement6 h l c h rmst be mat for sensitivity, stability, and cgltering to make any system rat i r fmtary in r&oe. The abi l i ty of the designer t o maat .those roquiruaatr depend. upon hlr undarrknding the effect of woh rymtsrp p.ramstw upon tb, roquirm d s . For amupla, to nuke a Judioious deoidon mgardlnq the magnituda md dlrtribution of fiiotion, the designer must &tormine the mulmum pumisrible frlct lon pattorn from his ertabllahod oentoring roq-mmnb, - nut det8rmlne by rnolysis or tat the mlnbmm M c t i o n pattarn ro@s~d t o arintaln stabill*, and then he our establirh the quality aontrol l l r i t r of m a t i o n dtbin eaah el-t of the qmtau. *

C e n t e r i n g charrctsrlrtlor M r e f o r d to harein ary be dofY.nod u the' rolaWonahip be- force, podtion, and r a t e of motion of a oontrol ~ I f r r in the rlcinity of the tadmned neutral position, Tho or i t ioal paiotr ordinarily oonai&nd rre u followst

1, Break-away f r i c t ion - defined a s the force (as measured a t control lever) required t o begin surface motion from neutral.

2. Centering force - defined .s the f e e l force v8. control position *en returning t o neutral.

3. Centering e r ro r - defined as the p o e i t i o a ~ l error or distance f roa neutral a t e c h control e t a l l a when being returned taward neutral by asr-c 'forces.

4. Centering speed - defined a s the velocity vs. position of control lever when being returned toward neutral (from various starting posi t iom) by aerodynamic forces.

When a power boosted surface control systerm is used i n an airplane tho aerodynnmic hinge 'momenta on the aurface a re approxiatataly d i rec t ly propor- t i ona l t o rmrface deflection and thus taper down t o sero hinge moment i n the trimmed neutral position. A f ract ion of theee hinge lpomonts must reac t in reverse through the control system t o produae f e e l force a t the p i lo t ' s control lever and t o return o r center the mechanism (when displaced), to the neutral position. Therefore, the degree of raversib1lit.y of the mschanim dotsrminss the centering characteristice,

Figure 1 i l l u s t r a t e s the boeic elements and t he r ignif laant exterrul forces acting upon a power boosted contzol system, The basla arrangemat consis ts of a W e c t manual drive h m p i lo t to ourface with a boost packqp inserted a t one point in the mechanism. The boost peckage c o n t d m t b s d i f f e ren t i a l linkage, actuator, actuator disconnect, control valve, and hydraulic accessories.

When w l y s i n g centering oharu)taristice, tho dosigner is rpeai i io . l ly concerned with six relatad v ~ a b l e s :

1. Control foroe o r p i l o t o f fo r t ( F ~ )

2, Control surface force (Fs)

3. Control valve force 0"v) 4. Actuator force

5. Surface position

6. Surface rate and direct ion of motion,

The magnitude of a o o o l m t i o n forces throughout tho mch.niem h s boon neg l ig ibh i n centaring studios t o &to, whiah gnatly rimpUfior tha arnlysia by e U a h a t i o a of aooeleration a s a variable.

Since pilqt effort and s*face force may become independent variables in a centaring problem, it is desirable t o expees each other variable as^ a function of these two. One soUd link i n the boost package serves as a foIlow-up loop and also measurhs the error signal. The four forces acting upon this link are the f i r s t four variables mentioned above. Taking e\moration of momnts on this Unk about the piston rod bearing and solving for FV ilPdicates thnt valve forces a m a direct function of control force and surface fome only as dotenninod by the goomtry of the control mechanlsn a t neutral and as expressed by

Taking summation of moments about the valve rod bearing which eUminata~ Ptf Md eoloiag fo r FA indicates tht actuator forces are olso a meet goometrical function of control force and .surface force only M eqwosmd by

To aid vimallsation of this problem, a graphical llrsthod has boen oployed. (9.0 Fig. 2.) Tho independsnt variables are control fomw plotted as ordinates, and aurface forces plotted as absairsa on rscturgulrr o o o ~ a t e s , By substitution of various constant valuer for Pg in the second equation at the top of FQ. 2, valve faroos in pounds are plottoti a8 a family of p r ra l l r l lines with scales along the sides of the chart. By substitution of various c o ~ t a n t values of FA in the f i r s t .quation, a o k u h r forces i n pounds are plotted as a fPrnily of parallel Unes xLth a rcalo .low the top of the chart.

Since ourfaoe position i r appro;ldkately a linear function of surfaao load8 it 8 b o asy be measured Jong the 8bscissa by using a scale factor (K) a h i s d e t d e d from the oir speed i n a given problem, 6 = C v2.

If q v two variables are known8 a aystsla condition is establlrrhui oo 8 pod& on this chart. A value for each other &able om then bo obtPined A.om the cbut .

Surface rate and direction of motion nay be detenalned by applying the valve load8 obtained irOll Figure 2, to the characteristic8 of . tho valve.

To proceed further with the analysis, reliable d a b must be obtained concerning the operating frict ion and loads for each eloment of the ayatAP. Figure 3 b a plot of laboratory test data obtained from a m i c a bnlrnned piston boost cylindar. During frequent intelnittent duty, the stall ond atarbing frict ion piston fome in pounds, indicated by the lower solid Ila6 on Fig. 3, can be exp~esaed arr

FA ' 12 + .OU x internal pressure (psi)

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The centering characteristics except centering speed are determined, i n t h i s case, f rom the f r i c t ion of the system elements. Break-away force indicated by poirrts (1C) and (16), is simply a combination of cylinder and control friction. The centering poaitional error a t point (6) is the intersection of the 12 lb. cylinder f r i c t ion l ine and the 2 lb. conk03 f r i c t i o n Une.

Figure 6 i l l u s t r a t e s a surface deflection cycle i n a boost syatem . employing a balanced s l ide valve with no centering spring. The operating load of this valve consists of 2.5 lbs. constant f r i c t ion a s indicated by b w slope diagonal l i ne of Fig. 6. Starting again frcm point (I) , equilib- rim i n neutral, a slowly applied control force proceeds vert ical ly to (2)

b where valve f r i c t ion f i r s t is overcome. The valve dl1 direc t presume to the c;ylinder proceeding along t h e 2.5 lb. valve action l i n e u n t i l control force i e no longer increased. The system will continue to move, proceeding horizontally t o point (3) where the valve readjusts toward neutral Pnd the system a t o l l s a t point (3). With a slow decrease i n control force, the valve dll immediately readjust and assist (as demanded) the return of surface to neutral proceeding along valve action l i n e u n t i l the control force reduces to 2 lb. forward control f r ic t ion (0 force a t pi lot) . Tbs system w i l l continue across the 2 lb. l ine to point (2) where a reverse valve load wi l l bring the valve t o neutral and stall the mechanism. The outer loop p o b t s (4), ( 5 ) , (6) and (1) are again the path of control follees a t the p i l o t and was constructed by adding or subtracting forward control f r i c t ion f m m the previously determined points.

Starting a t point (3), i f the control load i s rapidly released, the valve load w i l l exceed 2.5 lb., thus fu l ly opening the valve and centering the mechanism a t f'ull flow rate . The values of valve f r i c t ion and forward controls f r i c t ion selected resul t i n zero positional error. However, the analysis waa simplified thus f a r by neglecting f r i c t ion i n the driven l i n b g e .

Figure 7 i l l u s t r a t e s a surface deflection cycle similar t o Figure 6 w i t h the valve friction increased to 3.3 lb ., a control f r i c t ion of 2 Ib., and u i t h a driven linkage f r i c t ion of 10 lb. introduced. The break-awry force 18 b.6 lb. With this system condition the centering s t a l l point (4) oocurs a t -10 lb. surface load meesured a t the boost package. Diamond shaped points ( 9 ) , (10) and (11) a re surface loads a s measured a t the surface. These points are offset horizontally by + 10 lb., which i s the driven linkage fzLction from the oorresponding system co-ndition points. P o h t (11) indicates t ha t the actual surface load is zem a t the mechanism s t a l l point and thus zero canter-

" lug positional error is. achieved. Thia is accomplished, a s i l l u s t r a t ed in the upper l e f t corner of Figure 7, by establishing optimum valve f r i c t ion f'rom the intersection point of the controls f r ic t ion and driven linkage friction.

* T h i s tzLck of power operation to a perfectly centered position i s

damndent upon cer ta in qualifications:

1. No centering spring within control valve.

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When the centering spring is r-ved iron this open-center valve, t h speed cbpracteristica cannot be represented by a line. Rapid release of control force results i n centering epee& located a t the broken l lne for a closed-center valve without spring (and t h i s Includes stall point location). Slow release of control force results i n centering speeds loc8t.d at the Iln, for open-center valve dth sprlng.

CONCLUSIONS 3

In the pant, centeriag chulpctaristios of mroy power boostad m f a a e corrtrol a y s t a ~ ~ have b o a evaluated only by pilot eeti.Otion late i n the f l lght t ea t program of the alrplatm. Prlor to this point in the l i f e of the project, M X U ~ cmpromises have made to faci l i t8 to Isuwirature and to m e t other performeace requirements, mrch as pomr and stability, d t b no analysis of their Pdtarw effect upon c e n t e r i q characterlstlcs. At this point in the l i f e of the project, lo thing but rFrror detal l changes cur be made to improve performance without excessive penalties of oost and d e w .

T h i s papar proposw tht oentering characteristics be oo~ lda rod 8 baalo design problem t o be ~ o l y s e d early i n the engineering phaue ,of eaoh projscrt Md tO be permed by t e s t Md by quallty control throughout the e m projeot Ufe. To i l lus t ra te h w this can be mcompliahed, cent* characteristics have been dofined in abp le englnwrlng toms, 8 technique of amlysis has bean preasnted, a tes t pmc-e bas been ~utllned, Pnd appYcation of the analysis and test to four different systas vP118tlons hra been i l lustr .kd.

HELICOPTER POWERED CONTROL SYSTEM PROBLEMS

WALTER GERBTENBERGER

of

lPikorrky Aircraft Divirion United Airc raft Corporation

Bridgeport, Connecticut

HELICOPTER POWERED CONTROL SYSTEM PROBLEMS

INTRODUCTION

Since the audience comprises mostly "fixed wing1' engineers, it is f e l t t h a t some introductory remarks about the operation of the helicopter, as re la ted t o power controls, might be i n order. The type of control for maintaining a pitch and r o l l a t t i tude has gradually boiled down t o a single type. It i s a nethod of t i l t i n g the rotor thrus t vector by t i l t i n g the t i p path plane of the rotor. It i s a reasonable assumption, f o r a control discussion, t ha t the ro tor thrus t remains essent ial ly perpendicular to the t i p path plane. Hence t o go e i ther forward or sidewards, it is merely necessary to tilt the thrus t vector in the desired direction. Although a rotor may be mounted on a Gknbal jo in t and rotated by brute force i n the desired direction, the more prac t ica l way of accanplishing t h i s is to hinge tne blades a t t ne i r roots so tha t they may feather and flap; then applying an osc i l la t ing change i n blade angle a t exactly one times rotat ive speed, &ich w i l l excite a one times rotat ive speed flapping of the blade@. A n observer on the ground would see a constant tilt of the t i p path plans ra ther than the one times rotat ive speed flapping re la t ive t o the rotat ing drive shaf t of the rotor. This type of control is desirable, because the low i n e r t i a of the blades about t he i r feathering &s and tne smaller lnanenta of the blades about the feathering ~s re su l t i n smaller control forces necessary t o be overcome by the p i l o t through h i s control s t ick. The feathering i s accomplished by a simple swash p la te on a t ransfer bearing d i rec t ly under the rotor hub. Vertical control is obtained by up and down motion of the swash plate as contrasted t o the tilt required fo r azknuth control, and an increase i n the thrus t vector resu l t s from an increase in the blade angle of all the blades.

Before describing i n more de ta i l some of the problems of the azimuth control system, a few introductory remarks w i l l be made concerning the direct ional control. The type of direct ional control of the helicopter depends largely on i ts basic configuration, tha t i s , whether the helicopter is a single rotor helicopter o r a multi-rotor helicopter. The single rotor helicopter i s more analogous to the present day a m l a n e except t ha t the ta i l surface i s instead a small rotor to counteract the aerodynamic torque acting on the main rotor and to provide f o r direct ional Control above or below the thrus t required to oppose the torque. The t a i l rotor thrust i s vericd by changing the blade angle of the t a i l rotor. I n multi-rotor helicopters direct ional control can be varied by controlling the torque dis tr ibut ion between two counter rotat ing rotors or by t i l t i n g two rotors, so that the horirontal projections of the thrust vectors form a couple to oppose the sum of the torques applied to each individual rotor. The foregoing statements apply t o mechanically driven rotors. I f the rotor i s j e t driven a t the t i p of the blade there is no torque compensation required, but some meana of direct ional control must be available e i ther in the form of , a t a i l surface operating in the s l i p stream of the rotor, a variable, horizontal j e t th rus t a t the t a i l , an a z d l i a r y rotor or a combination of these devices.

Returning t o the discussion of the azimuth control where the t i p path plane is t i l t e d by means of the tilt of the swash plate, we can appreciate the problems resul t ing from the f ollotdng considerations. The f r i c t ion resul t ing from the feathering bearings, which are required t o support a centrifugal force of a p p r o h a t e l y 5,000 lbs. i s appreciable. The need fo r a f ine balance i s necessary between the aerodynamic thrus t on each blade and the centrifugal force component projected along the line of thrus t which prevents the rotor from llconinglt too much. Also, the

. variety of vibratory and gust disturbances transmitted into the control system from the aerodynamic surfaces which support the fulJ. weight of the helicopter and which move as a complicated piece of machinery miqfit be d

disconcerting. On top of t h i s a helicopter i s inherently unstable in hovering having an unstable mode with a period of the order of magnitude of 10 seconds o r more. m i l e the p i lo t can cope with this unstable osci l la- t ion because of the long period, it i s desirable tha t he make frequent corrections without undue effort . The problem was not insurmountable on helicopters of the 5,000 lbs. p o s e weight category, but did l i m i t the s i z e of the helicopter unless suitable potjer controls were developed. Various aerodynamic and gyroscopic devices have been incorporated on hellcopters. . Three of part icular i n t e re s t are the Bell s tab i l iqer bar, the Hi l le r control ro tor and the Kaman blade tab. The Bell s tab i l iz ing bar i s a large r a t e gyro rotat ing a t rotor speed ( i t s undamped natural frequency i e hence one times rotor speed) with viscous damping, which can be adjusted to o p t M z e the s tab i l iza t ion obtainable with a raze gyro alone. The Hi l le r control rotor is a s imilar device which uses the blades of the control rotor t o obtain aerodynamic damping. Another feature of this control rotor i s t h a t the feathering produces flapping on the small control rotor which introduces feathering on the la rge rotor. I n short, there i s an extra stage of an a e r o d p a d c .servo in the control loop. The Kainan rotor, the tab i s located on the blade near the t ip . A preloaded cable control regulates the angle of the tab which introduces a twisting moment on the blade, which in turn twists the very f lex ib le blade t o the desired angle without the use of featheflng bearings. *Each of these devices solve the problem i n part, but have cer ta in disadvantages a s well. Power requirements f o r aerodynamic actuated torques should be considered, and t e s t s run a t Sikorsky Aircraft with t ab controlled rotors wlth featherlne bearings indicated tha t the additional drag, due t o t h i s type of aerodynamic servo, resulted i n power loss f ran 5 t o 1% of total power absorbed by the rotor.

The use of hydraulic power appeared t o be the most economical method of *

improving the control of the helicopter. I n 1949 a hydraulic p m r control fo r the three inputs of the swash plate was ins ta l led on the S-51 model helicopter (gross wei@t 5500 lbs.) The power control had i n f i n i t e booet Q ra t io , that i s no feedback from the output t o the input, operated a t a pressure of 800 to 1,000 p s i obtained from a constant displacement pump and unloading valve charging an accumulator. The awragle power requirement for t h i s system was pract ical ly nil, being l e s s than a quarter horeepaser. Although the original purpoeb of t he control was t o prove tha t t he hellcopter could be f lown with t h i s type of control in order t o pave the w a y f o r the

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loaded by-pass valve closes, and freedom f o r the valve of the emergency system to operate is introduced simultaneously, so that the emergency system, which was previously idl ing in the control aystem, assumes i t s burden before a q of the control forces are allowed to reach the p i lo t ' s stick. There i s an automatic check by the p i l o t a t every start, because the eng5n.a must. be s ta r ted before the rotor, and during the i n t e r n between engine s ta r t ing and rotor olutch engagement the emergency system i s the only power control which can function a t t ha t time. I t is true tha t engine o i l is not a very good f l u i d f o r a hydraulic system, especially from the standpoint of high Piscousity a t low temperature, but with lagging on the lines and with a large by-pass located dlrect ly in the actuating cylinder, satlafactory cold mather operation can be obtained. The u t i l i za t ion of the casting engine o i l system makes the emergency aystem a convenience which i s feasible from an eoonomio standpoint and weight standpoint, since only an extra aotuator plus oonneotlng lines are required. There s t i l l romoins the undosirable oondition of having a oyllnder wlth l1O1l rings with t h e i r inherent f r io t lon wnneotod direot ly in .tihe control system during mrPrrl operation, and of for t r are baing ma& to roduoe this f r ic t ion , although the fome in tho r t iok i r rpproxhak ly 1 pound.

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The des i rab i l i ty of the spring force gradient was pa r t i a l ly off-set by the need f o r trimming t h i s force a f t e r changing t o a new flight condition. This configuration may not be important i n fixed wing a i rc raf t , but the helicopter is more apt t o be operating over a variety of f l i g h t conditions and there w i l l be more of a tendency f o r the p i l o t t o forego the trim changes and cope with the bungee force i n the trimned position. This does not mean he will not camplain about the d i f f icu l ty of flylns in these conditions. I n fac t , complaints of a trimnable bungee on the rudder pedals of the S-5s type helicopter has resulted i n the developnent of a paver control f o r the t a i l rudder even though it was not considered necessary af te r the i n i t i a l f l i g n t tests of tne helicopter. After f l i g h t tes t ing the first rudder servo with no f e e l it was evident t h a t the rudder pedals were en t i re ly too l i g h t and they would s t i l l be too l i g h t even at'ter 10 or 50 hours of f lying with them. A pneumatic cylinder was attached t o the pedal control with a bleed across the piston so t h a t temporary centering could be obtained, but a f t e r a time the pedal force was independent of pedal position. It must be remembered tha t there i s an appreciable difference in pedal position between climb and descent because of Wxerences in rotor torque f o r these conditions, which have t o be compensated f o r by the t a i l rotor. The r a t e f e e l of the prleumafic cylinder invokes no objections but the temporrry position f e e l did. A s a resu l t , the type of f ee l device proposed f o r t h i s control was me which consisted of centering springs across the valve of the power control so tha t a given r a t e of servo t rave l would be produced by a given force from the pi lot .

THE OVERALL FLIGHT CONTROL SYSm - - -. .-

The ar t i t ' i c ia l f e e l device should be integrated i n fhe overall control system t o a id the p i l o t in f lying the aircraf t . It supplements the p i lo t ' s vision and should. not transmit any confusing signals. I n f ac t it is believed tha t the f e e l requirements f o r any a i r c r a f t may vary considerably depending upon what is required t o x l y tha t a i r c ra f t most proficiently. I n the case of a helicopter, assuming t h a t good flying quali t ies a t hi& speed have been obtained with properly designed aeromamic surfaces, hovering s t a b i l i t y should be approved. A t the present time it i s a question &ether helicopters will become big enough t o afford tne added weight of a sui table auto-pilot, o r whether a suff icient ly l i g h t weight auto-pilot can be developed to i n s t a l l in the present helicopters. SikorsQ Aircraft is engaged in a program to attempt the l a t t e r . A s f l i g h t t e s t s have not been made on the initial proposal it is d i f f icu l t to say whether the auto-pilot w i l l be successful; but it is of the d i f f e ren t i a l input type so tha t the p i l o t may f l y with the s tab i l iz ing ef fec t of the auto-pilot always present. The system is integrated with the exist ing hydraulic power control units, which makes it possible to keep weight a t a minimum. Several types of f e e l are being considered including a type where the actual gyro signals convey, by means of a force, what the auto-pilot considers a proper way t o f l y a helicopter. There is certainly mamy problems connected with such a device which pending f l i g h t tests w i l l reveal and lead t o hoped-for solutions.

FUGET RESEARCH ON FORCE WHEEL CONTROL

by DUNSTAN GRAHAM, Lt., USAF RICHARD C. LATHROP, Capt., USAF

of Wright Air Development Center,

Wright-Patterson Air Force Base, O i o

In order to sccomplirh precieion flying, etatic a d dpnaaio rta- b i l i ty of the sirplane'fi flight path i e required. The prorlrion of tbir r tabili ty need not intorfore with the pilot 1 r control of tho flight path. Bbroe t ranrdwrr connected to the pilot 's cockpit oon- t rolr oan be wed to biar the referencor of the rtabiliring eyetem. This romultr in easy aontrol by the pilot of f l i a t whioh i s a r t i f i - cially rtabilired and coor0inated.

Tho practical mccerr of Neb a war demonrtrated on the A l l W e a t h u Branah's #ore@ Wheel B-17. Qpeeiment r were made i n the atateerr l lke a caras natural handliag q ~ u l i t i e t ~ and nonlimar force gradient configurationr.

A l l Woathor Branohla future rerearch plan8 oom&rire experiment8 with the C+ (dercribed in anothor pcqpor), and the application of force uhwd control t o other airframe antomatic control combination in an atternpt t o meet spcrcific operational requirmentr.

The military or economic value of an airplane depends on i t s abil- i t y to traverse a controllable fl ight path. It i s , often neoerurrp or desirable that the fl ight path o r certain reotionr of it be t raver rd with great precision. Conriderations of enroute and terminal ares navigation, landing approach and lending, interception and pursuit, formation flying, and bombing m a , i l las t ra ter the point.

WIPdammtally, the a i m a m i n flight i r a velocity vector i n space. It has a direction in which i t i s going and a mpeeb with which i t i s going there. The time integral of t h i n velocity vector i s the flight path. (If instantaneous velocities are nmltiplied by l i t t l e bi ts of time to get l i t t l e bits of distance travelled, i n the direction of the velocity vector, and these are rtnrrrg end t o end, the result i r the airplane88 fl ight path.) When the velocity vector i e steady the flight path i s nstraightn, and when the vector i r dunging the parti- ciple flmaneuverfngUescribes the flight path. The wse with whiah r pilot may f l y rteadily i e a Function of how w e l l the airplane r e r i r t r changes i n i t r velocity reator, i.e., i t s f l ight path rtabilfty; and the errre with which a pilot manavers i r a Function of the means available t o him to al ter the magnitude and direction of the velocity vector, 1.0.~ control.

Flight path etability and control has reveral faoes. The f i r s t af these to ehov iteelf was s ta t ic 'rtabllfty vith reepect to the relative h d . Leonardo da Vinci convinced himrelf that the center of gravity of a flying machine ehould be ahead of the center of wind resirtame. HIS ngreat birdn of 1506 was desieed with tr longi tudinsl s ta t ic margin of more than f i f t y percent chord. &host without exception, rince the Wright brothers 1902 glider, maceseful airoraft have had both longitudinal and directional r ta t ic rtabil i ty with respeat t o the relative wind.

The Wright brothare devoted considerable attention to w h a t they called nbalancing. The 1902 glider was their f i r s t maahine whiah imoqorated a f i n a d rudder. Together d t h their predoas dwelop- m a t of the elevator for lo~gi tu~¶ine l control, wing warping for ro l l control, and finally a power plant, it made flying possible. But "balancing" i 8 just what the f i r a t f l ights were, and flying ha8 r m i n d to a large extent a belending act.

The shadowy face of flight path stabil i ty and control has been . attitude stability. Attitude rtabil i ty cannot be inherent in an air- freme. By manlpulation of the controls a ekillful pilot can provide attitude sta3ilieation vith reepect to a natural or a r t i f ic ia l hori- zon and a direction indication. Thie l a , however, a feat. Bortunetely, an automatic pilot can be built to do the eame thing more rapidly and

precisely. The f i r a t demonstration of automatic p i lo t control of airplane at t i tude war made by Lawrence Sperry i n 1913. Note that t h i n "nboerdn etabiliration, by the p i l c t or automatic p i lo t , de-aende on control. It i e contrasted with %~tbooard"tabili~ation (8uch ae l a provided by fixed aerodynamic uurfacee) which i n t d f e r e e with control.

The i n c q a t i b i l i t y cf outboard etabilization with control ha8 fortered the opinion that a tabi l i ty i e achieped only by racrificing control. That this i e not neceeearily the cane may be i l luetrated by conridering the example of a rlmple poeitioning eervomechaniem, (Figure 1). The f i r s t desim i e deficient i n dynamic etability. The output shaft reproduore the aagle of the input ahaft only a f t e r a wdr ly damped traneient har died out. The designer hae a t leaet two alternatiree. He may add mechanical dampiog to the output ahaft, %utboardn. T h i n waetee control power and opposes rapid and accurate control of the output ahaft. (Such a ryetam has a eteady etate error d e n eubjeated to a velocity input.) On the other hand the deeigner may, Elnd usually doer, chooee to employ the derivative of error eignal to etabil ize the eyetan. UInboardn etabilization, accoqpSirhed at the e i m d level. doee not me te control power, and doer not reeult i n a eteady e ta te velocity error. In th ie cam etabi l i ty doee not interfere with control. If anything, control of the output ahaft hae been en- hawed and refined by additional etability.

The aonaqt , i l l u e t r a t a l by t h i s eimple example of e ~ t a n e o u s e tabi l i ty a h control, can be carried over into the airplane handling qualitiee f ield. Note, however, that both the e tabi l i ty and the con- t ro l shown here depend upon feedback.

Flight path etabil i ty, compoeed of at t i tude etabil i ty euperim- pored on re la t ive uind etability, i e commonly achieved by the feedback mechanisms known ae automatic pilote. These may be equipad with pedeetal controllere vhich provide a meane of introducing control a t the e ignd level. . Feedback for control purpoeee i e through the hurmrn pi lo t 1 s vioual referenore and. hle kineathetic emee fo r m a l l finger and wriet moveplente.

We portulate that a euparior form of feedback to the M a n pi lo t i e force. There i e coneiderable logic and erne cuperimentd. widonce to eubetantiate th ie contention.

Aeauming -that the fee l of the controls i s .important to f l ight path control and that compliance with the handling quali t ies q w i - ficatione inearee adequate fee l characterietice, but poeeible only marginal f l igh t path etabil i ty chsracterietice, i t i s logical to retain the f 'mi l i a r cockpit.controle'and their f ee l and add flight path etability. This i s the equivalent of eaying that i t i e logical

STABILIZATION OF A POSlTlONlNO SERVO

LOAD

LOAD

8; 8,

OUTB BOAR^ STABILIZATION

0; € , AMPLIFIER . MOTOR

+

POSITIONING SERVO DEFICIENT IN STABILITY

€3; E I

.

*INBOARD* STABILIZATION

F I b l m 1

LEAD

NETWORK AMPLIFIERr

+

MOTOR - 80

to hare thq coakpit controle command a control eyetern in tau& a way that the forcer epplied a t the controlr -produce predictable and pry- chologiocilly acorrecta airplane reeponree while the automatic control eyrtam ar t i f ic ia l ly rtabllirer and coordinate8 the flight. It may, however, be poeeible o r evsn deeirable t o take certain l ibertiee vlth the normal or natural reeponeee t o control forcee applied a t the cockpit controlr.

In general, the natuml reoponree are ae follows:

novator foroe oommende a proportional normal acceleration (Vg") o r a proprtional change from the trim airrpeed.

Aileron foroe commendr a proportional wing t i p helix angle (roll rate a t a given drrpeod).

Rudder f oroe comnmadr a proportional ddeelip.

One mn't add ormger and appler, however, and the rchge i n which foroe i r meamred directly and i e ueed to cornmead an ar t i f ic ia l r tabil i ty rystem to probrrce natural rerponler require8 that the primary referen- oer for the rtabiliriog ryrtem be the f l ight variable8 tmich i t i r &aired to command.

No nruh ryrtem yet d r t r , but i n tho f a l l of 1950, a t the behest of the l a t e W o r Patriok L. Kolly, the A l l Weather Flying Mvidon undertook to mponaor the dwolopment of 8uch a rtabiliring ryotem a t the Oornell Aeroneptioal Lsboratory. Thir ryetem ham been described in another --or.

A t the mame time a d l group under the energetic leaderrhip of Major Kelly beg= to conrtnrct a jury a r t i f ic ia l r tabil i ty eyetsm vlth force commandr, ueing ounponentr which were readily aveilable. Right uring automatic rtability and force control wae f i r e t achiwed by A l l Weather m n g Divirion on 5 Maroh 1951. Throughout the ~ m m e r and. f a l l of that yaw the automatic rtabil i ty and force control rystem war terted in reveral oonfigurationr. I t i r the rtory of thore ex- perimentr that i r the principal mbj ect of th i r paper. No rocordr of t e r t r rerultr were ever made. We out and we tried and cut -and tried again. Our goal war an airplane that had the feature8 of the ryrtm-- to ehow that i t o d d be mabe to work and that pilot8 did l lke it. I t s operational auccere war to hare been meaaured later.' A l l Weather Plying Sectionlr rerearch on force control var unfortunately out short on 7 ITormber 1951 when the tor t airplane crashed.

40 r ta r t wlth there war an eirplane with an automatic pilot inrtalled. !The force control tmb-ryrtm w i l l probably find i t 8 prin- o ipd uraiulnerr i n a l r & f t in which r p a e and weight are a t a prm- ium and the pilot murt be raliod on t o do a r much ar he can. The

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conrtsnt speed). While th i s ryrtem worked and the pilots experi- enad no difficulty i n flying it, they were unanimous in condamming it. Unfortunately no one flew it who had had no ezperience in aa airplan.. TUr might have shorn 'nor natural a method of control it i n .

The control position feedback was unnaturally related to the force f e e d b e . The wheel--rigidly connected to the ailerons-followed th- i n emteblimhbg a bank. Ar every pi lot knows, the no& requeme of wento i s romething l ike this:

1. The whed i s rotated into the turn.

2. It i s held i n a deflected position.

3 Then it ' i n returned toward neutral.

4. Finally i t i s slightly rotated oat of the turn and held there to counteraat the overbanking tendell~y.

With th i s f i r s t 8yst.m the automatic pilot did all tni r while 'tlu pilot maintained a conrtant aileron force. It 's almort iapoeeible to beliwe, but neverthderr tme: not only way control smooth and stray, bat nobow objected to this peculiar porition feedback. The objectionr ware aoncerned with the magnitude of tbe forcer required to maintain tb. aircraft i n a steady tm. There objectiona, might have been wer- come by using wa.hout or nonlinear gradientr or both. Brief eqsri- manta ware conducted with a strong centering characteristic and a low rtiuk force gradlent for mod'arate diaplacementr. Thir artearr l ike a carn reanr of control, with or without nonlinear gradientr, howww, did not meet with agproval and was qulckly abandoned.

The next configuration which was tried war designed as a f i r s t step toward making the aircraft control characteristics correspond to those of a conventional aircrait. Specifically, the ro l l control channel war alt sred in such a way that whenemor an aileron f orce g r a t e r than three pounds war applied, the vertical gyro r o l l signal and vertioal gpro ro l l erection were removed, and, by means of a ratchet relay, raalned dilrconnected unti l a button on the control wheel war deprersed.

The mdder channel was modified as followr: the directional gyro rignal was completely removed and a lateral accelerometer was rubsti- tuted ar a yau reference device. !Chis configuration reeulted in a control characteristic i n which wheel force i n ro l l commaaded an ai l - eron displacement (and therefore rate of rol l a t a given airepeed)

and rudder foroe commended lateral acceleration (and therefore rido- d i p angle, epprox5mately). Thn 'action of the lateral acceleometar v a e such a r to maintain negligible aidedip i n the absence of a rudder force m i g a a l , 80 that coordinatod turnr were made uripg the vheel alone. This rimplified the control of the efrcraft in nornal man- cnvarr. Pilotr frequently ure l i t t l e or no rudder in normel man- c u t e r e , particularly i n nailti-e~gin6 or high-.peed drcra i t . The rudder pedal force pickupr, hovewar, dlo- the pilot to inten- tionally r idedip the aircraft i f he eo deeired. (~ntslrtiond. ride- d i p i r u r d in making crone vird landingr, for example.)

No modification8 were made in the elevator control channel; the control characterirtic in vhich wheel force conumnded pitch angle w r retained.

Phe r o q m e of event8 in making a turn vith thir ayrtam war romethipg l ike t h i n :

1. The pilot applied a rolling force to the vheel. For very -1 uhed forces--muah mailer than thore o r dinarily usad in rolling into a t&n--wheel. force con=nded aircraft bank angle. When the vheel foroe reached a value of about 3 poundr, the tart ical g r o ro l l rignal war smoothly but rapidly (Umonnected from the automatic pilot, and rol l erection in the rar t ical gyro war 'disabled. A t thir point, wheel force commaa- dad eilaron dirplacs~lent, and the force vhsel ryrtcm acted merely ar a control booat.

2. To maintain a conrtsnt turn, the pilot relaxdl the vhedt force &en the proper bank angle had bran reached. Coordination war aupplied automat i d l y . Winor correc- tione to bank angle had to be made in the uoual manner, since no ar t i f ic ia l r tabil i ty elgnalr vare provided.

3. To return. to straight and l w e l flight the pilot hat¶ two choicee. Ee oould push the but ton on the rrheel and relax uheel force, in vhich care the vertical gyro r o l l rignrrl and rol l erecrtion would be reengaged, end the aircrait v d d automatically return to lsrel, etabiliswl flight. Alternatively, the pilot could return to l m e l fl ight i n the conventional manner, and then reerrgage the vertical gyro etabilisation by bepramring the button, ii he 80 derirul.

Smsral f l ightr v e e ma40 vith thie eyrtsm, and pralirpinery re- d t r indicated that i t war a very workable and natural ryrtmn of edr- craft control. Some of the pilotr vsre gratifylrrglf enthuriartic about the Improved fl ight path atability a d the sere of fl- path

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control. Unfortunately, the t e s t a i r c ra f t wae loe t before ext ensive evaluation of the syetem could be etarted, and before intprovamentr could be incorporated which would have rendered the a r t i f i c i a l eta- b i l iza t ion operative at a l l times.

A similar but improved ins ta l la t ion i s currently being ma& in a c-54 type traneport a i r c ra f t a t W~ight-Pattereon Air %roe Baee. Unanswered reeeatah queetions concerning much equipment &re legzon, and the operational uses to which it may be put renain to 'be explored.

Coneider swera l possible force wheel configurations: The block diagram of an danemtary ,force wheel system for one a i r c r d t a r l e i s shorn in Figure '2. This arrwement operatee a$ a boort only, and no a r t i f i c i a l e tabi l i ty i s provide&. The wheel i e r ig id ly connected t o the control surface through the normal control cablee. Thie l ink , however, doer not normally supply an appreciable portion 'of the hinge moment, but merely ac ts a s a rurface poeition feedbaak t o the p i lo t . Syrtems'of this t m e would tend t o make the force-biglacement gradient of the control wheel re la t ive ly independent of airweed. Thie was the c h a r a c t e i s t i c of tho l a t e r B-17 inetellation.

A rormewhnt more refined mystem i s ahown i n block diagram form i n Bigure 3. In t h i s arrangement, the rigmil from a eeneing element (ma& ar a rate gyro or eidesl ip indicator) i s fed back i n such a way a s to alter the effect ive dgnamiee of the aircraft . Presumably, the syetem i r deefg~ed so tha t the dynamic s t ab i l i t y of the a i r c ra f t i s improved. The r ig id l ink betveen the control surface and the control wheel i s retained. Thir means that control surface deflectione due to both wheel force signals and art- i f i c i a l e t ab i l i t y eignale m e ref lected in deflectione of the p i lo t '8 controlr. In earno caree, t h i s may be undeeirable. The degree of undo- s i r ab i l i t y appear8 t o be a function of the magnitude and frequency content of the a r t i f i a i a l s t a b i l i t y signals.

I f i t i e neceeeary t o closely approximate normal a i rcraf t f e d char- sc ter ie t ice , aignale proportional t o airepeed and Q n force may be introduced to vary the gain of the force whed prsanplifier and thsrefore altar the f orce-dieplacement gradient of the p i lo t s controls.

Methode whereby the force wheel control eyetedu may be ueed with an attitude-type automatic p i lo t a r e shown i n Figure' 4. In t h i s configuration, a eignal proportional to the integral of wheel force i e fed into the automatic p i l o t amplifier i n such a way ae t o command a i r c r a f t attitude. l r l l of the s t ab i l i t y aeeocicrted with attitude-type of automatic p i lo t i e retained during all normal manavere. The control wheel . theref ore become8 a sort of refined pedestal controllere wlth nfeeln. The r ig l a l ink between the control surface and the control wheel is retained.

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The configuration described above r e m l t e i n a control char-ta- i s t i c i n which, e f fec t ive ly , wheel fo rce commande r a t e of change of a t t i t ude . If t h e in tegra to r i s removed, the lateral control ChBSac- t e r i r t i c becomer s imilar t o that of an automobile, in which wheel f o r ce comr~andr a r a t e of turn. Integrators f o r the B-17 i n r t e l l a t i o n were conrtructed and grand-cheeked but were never in r ta l l ed . Improved in tagra to re w i l l be a fea ture of the C-54 ins ta l l a t ion .

In Figure 5, the p i l o t 1s controls a r e not oonnected m e c h i c a l l y t o the con t ro l eurfece, and motionr of the control taurface induced by arti- f i c i a l a t a b i l i t y 8ignd.r from the rensing element a r e not a o c o a p d e d by r imi la r motionr of t he cockpit controle. This mean8 that the p i l o t i r not rubjected t o the poesibly disconcerting phenomenon of having the cont ro l wheel move about i n a manner a i c h he does not command. Ae far ar the p i l o t i r concerned, he i r f l y ing an a i r c r a f t which i r more atable, but just ar con t ro l lab le ar t h e rame aircraft would be i f i t had a more oonvsntional control eyrtcm.

The fo rce wheel eyetcm of Figure 5 l a a ro-callod #full-poweredn ryrtear, i n which none of the power necessary t o more the control rurfacer i r mupplied by the p i l o t . This i e i n contras t t o the systemr p r d o u r l y described, which a r e boost e y e t a r , eince a emall portion of the power neceerary t o move the control surfaces is a p p l i e d by the p i l o t through the r i g i d mechanical connection between t h e cockpit controlr and the control arrfacer.

In a f u l l powered ryetem there i s no conventional control #feeln f o r the p i l o t , ro i t i e der i rab le t o include a r t i f i c i a l f e e l i n the form of a bungee rprlng, and bobweight wlth proviaions f o r var ia t ion of t h e bungee charac te r ie t io r with airspeed. The e f fec t of t r i m may be rimrrlated by including m e a n . f o r varying the cockpit control pos i t ion at which no fo rce i r q e r i s n c e d . It may be ded rab l e , however, t o f i x the r t i c k and f l y with forcer alone. Thir war to have been t r i e d on the B-17. In thir case the re would be no control po r i t i on gradient and no dieplacement o f the r t i c k wlth varflng trim.

Another fo rce wheel eyrtem using full-powared control e+rfaoee i r ahom i n F igwe 6. Thir arrangement i r r imilar t o that j a e t dercribed, arcapt that the a r t i f i c i a l f e e l bungee has bean replaced by a wheel pos i t ion B e r m r y r tm . !Che force-displacement gradient be conveniently abjarted, and, i n f a c t , may be mede nonlinear i f desired. Honlimar gredisntr a r e mupporod t o be psychologioally optimum. Pror i r ions a r e &own f o r t he introduction of alrmpeed aad #gn fo r ce rignale i n to the force &eel preamp- l i f i e r t o eimultaneaurly alter t he control rurface and wheel fo rce gradientr . Thir i s e s sen t i a l l y t he aystam which i r t o be flown i n the w.

The force &eel r y r t m s mentioned above a r e only a fsw of the many raible configurations. Testa with s y s t a s similar t o those of Figurer

$and 6 rill be c o n d ~ t l d by the A l l Weather Branoh i n coming months.

The use of force vheel control, i n conJunction with inboard rta- b i l iza t ion of airframe dynamics, maker rimulteneous f l i gh t p t h s t a b i l i t y and control realizable. A l l the f l ex ib le methods of reroo- mechaniun loop compensation a r e placed a t the disposal of the a i rp lmo handling q u l i t i e s designer. While many research questions conoerirrg optimum s t a b i l i t y and control remein to be i n v e s t i g ~ t e d i n f l i gh t , the technical prac t icabi l i ty and enthusiastic p i l o t acceptance of force wheel control have already been demonstrated.

1. Graham, D. nglight Path Stabi l i ty and Control i n Relation t o All Weather blyipgn USdF MB WCT-2372 29 October 1951

2. Milliken, W. F. @Dynamic Stabi l i ty and Control Besearah, Section V I IIn Cornell Aeronautical Laboratory w o r t No. CLLL-39 (paper presented a t tho Third International Conference of the B. Ae. 9.-I. A. S. Brighton, IDDglsnd S e p t ~ l b e r 3-14 1951)

3. Perkins,C. D. and Graham, 9. tlControl Force Instrumentation For Flight Teetn Rinceton University Aeronautical Hbginearing Laboratory Report No. 107 30 March 1949

4. Orlanelry, J . nPsychological Aspects of St i& an8 Rudder Controle i n A i r c r a f t V e r o n a u t i c a l W i n e e r i n g Review vol. g no. 1 J a n w y 1 9 4 9

AN ANALYSIS OF FULLY-POWERED AIRCRAFT CONTROL SYSTEMS

Including Some Reference to Flutter Theory

D. T. McRUER G. E. CLICK J. W. HAGER

of

Northrop A i r c r a f t , Incorporated Hawthorne, G a l .

The rill assrtrol mtr im8lJS.d fwtum8 ap i n t e e rarmtd ~ v o - c y ~ e r oa@ination. - the peramterm b o M d ih the derAmtioM of the equatioM are: ooppnsribWty of tho rbrkiag fluid, a l l n j o r o m t r olartieit iar, a l l apppmt sourear o i dmp*,'Md .od a o ~ c offact..

Comridmtiaai is glvm to tha affect of th, mjor mtr p r r u o t a n upon the darinsnt fmcpmq rodeo ami upam mtabillty. Pbioibla rodiiiaatiopr to tha clasoical f lu t te r t b m v a m a h qdlaatod,

Tha a u t b n wioh to aclaawledga the valuable aooiotcm~e and ouggaotiono offared by tha fo- indiridua108

T U e paper prorents an analysis of a typical valro-q-er hydratill6 actuator as ahoun in Pig , lo

Tho valve, being Integral with the cyliuder, proridear

- - r s r c n a v m u m ~ Gno 11- 18 1Me l L O n Ud ontrainad air will nwer conrtitute an aapreairble part of the norklng m1rrp.e of fluid4

2. Poaltion error ar a function of load, eamon to nap- contern ryetam, armmer negU&ble proportlaw bouue the neutral leakage (region of open eonter t p operrtion) ir confined to oxbrreb -l.l dirplaemta, (?%go 2,)

"W@J@tI pendm ao poqtmeard euofenfouoo eqq uo peouq oq mp8e.x mn eofqomd uZQoep qrq

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Tho plaa i r a8 follarr:

1. Dorive the flou q u a t i o p o

2, D m r i n the rilpUf$od -t(i lord eqg.tioa8,

3. Cambine the trro aqi dircurr the rirplliiod apt-,

4. Dirollrr tb M t b g carer 'of the rirpllfiod aystr.

5o Dorive the aquatiom for the g ~ e m l lord oar. i n c l d b g effect8 on f lut ter th .oqo

Tho tollawing ar.rptloxu w i l l gor.rn tb analpir :

2, A l l e l r a n t r of the r p t r can b. rqla8.d W Irsp.d COIL8wt8 o

30 The f l u in tho 8yllular i r elrap crprobrod t6 the aa tamt tht earitotion i r rragllgibleo

40 valve i r derlgmed in mch a way a8 t o nk. ne&lI@la the rate of o h q e of rorartr ef the fluid a r it p88@8 thmwgh the valve, t h r rerrrlt$ng in the all&amtioar of camtarh# and d e c e m t a ~ foreer on the valve,

5. The pirtoor l a double d o d o

60 The v a l r . i r m o t r i e a l , iOao, in i t r noutral poritioll a n noutral laakage a n a r .n .qPal.

7* Srrp p r e r m ir e r r a n t i a w sere.

mFL - The error, i , of tbir r e r r a . c k n i r ir the - d l f f emse bat- the 61- d i . p k c r a n t n k t i v e to fiuQ &rU8ttm, X, , ud tho 8yUlrd.r d i r p h c a n t m k t i v a to the m u m fiUQ & ~ 8 t ~ # X, IAt the pmrturbatisnr of the d i e p k o r a t r %,, . X , . and k!

Ooeo red em;Co* ey tg areqm

4 -A@ ~0-b 0% so epv -owd qaTq eqq o?w urn eqq -I eqq -@a esml-;C ;~arq-u eqq emeq- 4wqq ! a-k - a dm~uue~~ e.mooad eqq jo moq rq peeentdre fi-0 aim of af pa l a uodn eotmaprPsdep pw~qamy eq& 3 d~arqmu arj , quom~oqdefp

urn eq? pum 4uoqm~d eqq jo em0 uqqre uo l 3 pm eemsoad remPo eq3 & 4elmaee.td &ddao eqq jo aoyqomry e of romLa eqq e~m

0% bPF-3 eA'C1BI -3 mU UTWOJSe e1(& - UTVA SH& WWd WJId

4 = gravitational constant

ur = epecifi c weight of f luid

Ap- f-4 = pressure drop across or i f ice

Ptg , because of valve s)raetry f = *t

Inside the neutral leakage region the flow frolr the valve mt be expressed by more campllcated functions of the above mentioned variable#. However, the object of this short discussion is t o mnphasizo that tho flaw from the valve is predomfnantw dependent upon 4 , E , pad

4 , and that the relationship is continuous fo r a l l practical purposes, Therefore, since

the f i r s t order Taylor98 expansion of the perturbed flow becaror:

where , jc , and 9 4 are perturbation quantitieso The t e rn f l is a dlsturbaace entering front the p e r source, and need not be con- s ldqed further since this discussion deals on* with the surface actuating # ~ l t ( l ~ . a Therefore,

m i c a 1 variation8 of valve flow & with preswre differsnce & , valve displacarent E befig held constant, and of valve flow with ralvo dlsplacarent, pressure difference 4 being held constant, are sham In PI& 4, Figo 4 also shows a typical operating point, the o t w s t a t e values def inhg the operating point ( a!" , A* , and e*L ) pad possible values of the perturbed quantitieso The par t ia l derivatives of eqrration (b) are the slopes of the curves evaluated a t the opor@ting point, Note that on Fig. (b), the f i r s t quadrant shows the flow cod i t ion fo r a resisting load, and the second quadrant fo r an overbadbg load.

Variation of Valve Flcnr with Pressure Drop A C I P I ~ the Piston

(valve displaeeaent f raa neutral constant)

Variation of Val- Flor with V e l r . Diaplaccrat

froa Ueutnl (Proreme dlpp a a m r

piston conrtaat )

Fino 4

is a b p s positive o r sero ami tht a&/& 8 set of ~ t i o n r wtrich ar& almost alnap

a necessity for srstcn atabil i tr , It will be advantaeeoua in the dm-- rent that follows to maks the 81or~al siges of these qrumti)iiem appamt, that is,

Then equation (4a) mar be written

Hinas si~pl denote8 t Figo 4 (a) is a h y e

The term Cp is analogous t o the slope of the torque-speed curve of a shunt motor, and gives r i s e t o a similar damping action. The t e n C, is analogous t o the slope of the speed-field currmt curves of a shunt motor, and giver r i s e t o a similar gain texm i n the overall o p t r .

Cp varies f r m sen, t o inf in i ty a s a function of E and of 4 ; t h e s e m o c c u r w h ~ t h e v a l v e i s i n n e n t r a l w i t h ~ p n o s r u e d r o p across the p i ton, and the inf in i ty occurring a t the system s ta l l . r THE FIIlkl INTO THE CTLINDER - It now becomes necessary t o re la te the

. flaw from the valve with cylinder motion, piston motion, a d the pn88Ure drop across the piston, In developing the equations f o r cylinder f l# the canpressibility of the f luid w i l l be taken into account.

The t o t a l maes of f lu id on one side of the piston a t any ti.s t is :

where I = volme occupied by the mass

and p - f l u i d d m s i t y a t t h e t i m e

and the volumetric flow corresponding t o t h i s mass flow r a t e is

The compressibility of any f lu id is characterized by the exprwrlon,

where N = bulk modqlus of f lu id (force per unit area)

p - instantaneons density (mare per tmlt - h a )

P insbIltaneotl8 fluid pH88tWe

Dividing both sides of equation (10) by dt ,

(u) f 4-1 & dt N dt

and substi tutl~q'equation ( l l ) into (8)

and

where the subscripts, 1 and 2, denote the mglon6 on one and on the other side of the piston respectively (see Fie, 3).

If it i s asrnned that the instantaneous flaw into m e ride of the cyllnder is equal t o the instantaneous flow out of the other ride, 1.9..

and ii it is farther asuamed that the bulk modulus /V I6 conetaxit and identical f o r both regioae, equations (12a and b) be-

dY drr Since / r - - , it follaws tbt d d dt

'(a) o3w (57) pw ('K) eagqmbe BnF~n3~3oqw

ouoqeyd eqq eeatoo m3mueJ;rrP emseead eqq ' 9-2 zg

where %,, , %+ , f i , and A r' are perturbed q w t i t i @ r , aad st* 'd &* are steady-state values.

Slmpllfication of equation (17) may be obtained conaide- specific operating .,conditionso For eutample, since perturbatioarr are. be- considered about a steady-state cyllnder pres8u.m di ifemntir l , the tom d4 */df must be disregazdedo PPrthernore, it is a recognisd fact that hydreullc servos generally are nearest instabili ty a t , or nwr, the no-load conditiono* This zero laad condition closely c o r r e q a d r to a faired surfaces position, wherein the V O ~ U I B ~ B of o i l on either ride of the piston are very nearly equal. The effective o i l r o l m e bar beem previously defined as

4ra j &+G

and since A = -A - d then r l d k dr( 6 - c / ;

5+G

so i f 9 c, then fIr'=o ,

a constant bc I*

or

The approdmate linearized expression for perturbed f l a thw obtained under the foregoing conditions is

The physical representation of the o i l within the cyllnder a s a spring is developed as follows:

Consider, f i r s t , an open ended cylinder that bas bean subJectad to force d F and ccmpressed a U e a r displacement A A , as shown b, Pigo 50

* The effective damping tern, Cp , approaches eem a s 4 , ud hmce the cylinder had, dtninishee, See page 8.

Pram eqpatian (9)

and the volune incranant is 4 r = R (AX)

so tbat,

- qa iv (~1an t wring wn8t8nt of the oi l ,

Act- them is o i l in both sides of the pis- so that offeetively them are tm springs in parallel; hence the effective spring conatant is

I q & where 8 = - a s in oquation (16). G+4

TEE UllD EQUATIOHS

The true load upon the cylinder is mther inWd ami w i l l b. conriderad later. Bue to, its c 4 a p l d t y it is desirable to u8e a 8kpli i i .d mtr to obtain b r i c dera tanding a d than use the at- s p t r to dotelaino tho offeet of sa. of the less important elaarts upon tho mta operation. The sirplrriod sp tm is illuutrated kr Figs. 6 .rd 7. It w i l l ba noted tbat the piston rod is armmod rigid sld that tho vatvo displacaant 3?, is eanaidered identical to tho sono input 8-1 %z .

The error equation ( l ) , with X I * ir, R b-es

also, since the piston displacrnmt Xf is ngn-drtant, tho n9r quation (1Ba) becmes

The flow .quation8 (4b) (page 7 ) and (Ua') c a b h e to form the q r e r r i o n

The sipaplified load systm~ is rrav camplete3J defined aquationo (21) throggh (231,

It now becomes a rimple algebreic task to combine equatiOIU (21) throu& (23) into the o p a ~ loap transfor funetian of the r;ystr. definition, the open loop transfer fmction 3npUes that a l l artrmeou inpats, or dirturbances, are %em; hmce the aemdpamic feree F ef equation (18) is sera.

f Ewr valve neutral, vita 4 =O , the open tranrf er function becaes simple, since cF r O o (See F i g . 4.) Haling t M 8 abst i tut ion plus the nlationship 4 = A W ' r (the effective spring due ta o i l ~ r e s s i b i l l t y ) , equation (a) bemar:

Eqrrgtion (25), though representative of a rirpllfied mtem, is still quite q l e x . It i s therefore of considerable ralue to a t t r p t approximate factarieation, and t o consider Uniting case80

Equations (24) and (25) are of the forp

as- that the denabator frequencier are wldely separated, it is possible to obtain an'appmxbate factorisation of the donainator In eneral torrs0 (See Table I,) The mathod originally developed by Lin Refemc9 2) is enplopd for the factor is at ion^ it i s asmod tbt the f

f i r s t trial solution wnvdrges ianediately, In this situation, the Bode diagrerp of the r h p l U i 4 s y a t ~ l will appear as r h in Pig , 8, k- prlmental point8 are plotted os top of the calculated valuer for a ttpid p ~ i ~ s y 8 t ~ .

C C U P m ZERO - (Bulk Modulus infinite,) For the l i m i t - case in which the qmpressibility of the fluid beomes vem mal l , the laop transfer fuuction becanes:

h u a d the neutral position, with 5 = 0 , the open loop transfer fuuction reduces t o a very simple expression.

Equation (28) w i l l be recognized as the conventional expression used in the analpis of -radio amplifiers when used with mechanisms b a a large lags a t fmquancies mch lower than those of the wraulie amplifier. Epuation (28) is also recognized as that of perfect iategrator a d zem podt ion error servo, The reason for this is, of course, tbat. in thir care there are no spring or fr ict ion loads. I f the l@raulic amplifier i t se l f is ntable, t h i n approximation is an excellent one when the other elanants of a control loop ouch as an autopilot and airframe s e n e to lovrr the possible gain cmsaover frequency, & = cg /A . For une In such analpen, than, the hydraulic system is adequately dencrlbd am

-

The tima constant i s capable of being made quite mil with proper derQn, down t o the order of v100 second in the authorsf emprlmcen.

C ~ ~ I B I L I T T INFINITE (~ullc Ibiulus zero:) I f valve damping, Bw Tbetween the valve spool and the valve housing), is included an an extension t o the simplified load systeu, the open loop transfer function, equation (25). can be- nbrmr t o be

For the case with M i n i t e campressibility, the above open loap transfer functian became8

Input motion i s coupled to output motion only because of the riscons drag of the valve an the cylinder, The system is no longer a Gem position error device, Physically, the use of the relationship of equation (30) appears when two hydraulic ayatans are connected in parallel, with the power t o one system shut down, or on a single actuator installation with p e r off, If there is no viecaus frict ion caupllng between valve and cylinder, the transfer function beames zero, art them is no motion t r a n d t t e d fmm input t o output,

COUPLING SPRING -4 w - he situation occasio- mi r e s where backlash occurs between the cylinder and the surface load. It is then desirable t o investigate the s tabi l i ty of the systaa within the backlash region, though it must be remembered that s tabi l i ty inside the backlash region does not necessarily rule out the possibility of a limit cycle occurring due primarily t o the backlash,

he open loop transfer function with .A =a then becaer :

which, when Cp = o reduces t o

C a n p a r i n g equation (32) vith eqat ion (25), and also canparing \

the Bode charts of Figares 8 and 9, it i s apparent tbat s tabi l i ty of the lgdraulic ap t en outaide the backlash region assures s tabi l i ty inaide tb bsckLash regime

DAMPING COEFF'ICIWTS 4 .. . AND Brr 4 < Ce - Wh- the surface loading produces high hinge m ~ ~ e n t s , the pressure drop, , aomss the cylinder approaches the ap t an pressure, . This condition corresponds t o a very steep slope of the flaw-pressure curve (Fig. 4(b)), and, hence, the gradient Cp is verg large canpared t o a l l other s@an dompine.

The opem loop transfer function then becauest

Approxhats factori&ation of the danaatnator yields the undamped natural frequencies: .-

Note tha t d$, is identical t o k;', l i s t ed in Table I.

S P D G CONSTANTS AND MASSES EQUAL - A large portion of the foregoing discussion has t o be based upon the asmmption that approximate factorization of the transfer function d m a x b t o r i s ssible. This t a c i t l y aeeumes tha t the ad /-- are considerably separated, I f these quantities are wt separated by a wide margin, there a re no means available t o detezmine the effect of change of an individual parameter upon the system except by the use of numerical values.

It is, however, valuable t o investigate a Umiting case where the values of =/M,

' and are close together, the case f o r which Aa/, - 4 =-A' and M= = Ms = M . For simplification it i s also assumed that the valve damping, 8, , is zero. The open loop t r a n s f e ~ function then becomes:

uhlch, with approximate factors becanes:

The Bode diagram of this system i s shown in Fig. U),

EFFECT OF ADDRIG A SPEUNC IllbD AT THE SUBPACE (see Fig. 7.) - The mt i re s p t m amlysis to this point has separated the hydraulic 8pt.L frar eixteraal syataas a t the portion of th load schmatic where the laad F appears. me ruvierlying mason for this procedure was to prorido a maans of hwxllhg the f lut ter analysis of the hydraulic system. How- ever, in the operation of a Wraulic syatem under fUght conditiono, the force F wlll be tho major load, and a portion of this force may bo a definite function of ZS . Whem such a force d o t s , the force F ceases to be purely an external disturbance and became8 a p h q element of the inner loop. The sinplest example of this situation exists when F =As X , or a pun, spring load mch as a static hlnge mment gradimt is applled to the 8pteno In this went:

An approximate value of the naminal lawest &pal natural frequency is, with Cp r o o

Ilo-Uy As is much less tlnn 4 , so that the lowest undamped natural f reqamcy is essentially unchanged,

!CEE GEaaaAL mAD CASE

The purpose of this wction is to present the equations for the e u l l c o p t - with a l l of the terns taken into account, t o indicate the use of the lgdmullc syaten data in f lut ter calculations and to In- roetigate the effects of various stmctural caerponents upon the M e s t &pal natural. frqumcy of the open loop transfer function.

The laad equations, (40 throagh (43), are writtat frrrm Fig. I l (bat with C, and C, - Q as follmr: (See Fig. ll).

In addition to the above load equations, me further empmsrion is neceraaq for the am48ia of the oclplete h y d ~ ~ c - a e ~ c wtm:

This equation i n obtainad fna the relationships givon by equation8 (I), (4% (a).

The characteristic detembumt for the above array 18:

The outtmt 4 i s then

-tion (a) may be remitten,

in which case (46) becomes:

(The nprlmesw denote the new f o m of A, aft)r substitution of equation (4.44 for (44)

/ 5s. where = CE - 4

RecalJ3q that

Oesua pliq pe~m.dqs eqq roj suo~qdmnoea epq jo euo sw s~gq 2wqq mwae?~ i

71

It is of interest to anmine the lowest a roaPlate m d q m d natural frequanq of tho d. .o intor of oquation747a) W p i n a certain amount of insight into effects of the qutm o l m d 8 upon splrtr 8tabUty. Thir ia daw h the saae mmer as war doae p m o w l y vith

8hplIfi.d msot Th. -st tmbmpad natuxul frequan~~ ia appmximtoly htth& 811 88-8 negligible pr~port- tho

qpotiamt fly44 :

(48) ziresl C

AA-6

=, I

and uhm Ms >> MC d'%

(Uh) 4 j 2 = / 1: = -

uhora Jr is tho eerlea epivalant spring constant as eoea tho em- faae MM, MS .

Tha g a ~ a l a l e q u a t l o ~ should also be used in f l u t t e r talculationr, Tho tnrsl.ior function F/%= is sasily foxmad, sad u a d t o rophco the a w t a u y Inertla, damp-, atmi spring looking into tho &ace. Tho k c i t armmpth nrtrlcting this umgo is tht boding a d torsiosr do not eouplo with tho ~ m u l i c s t r o Beforring again t o oquatiasu (40) throrrgh (45) it i s apparart tbpt tho ruriaco motion may ba apmrrd

Since tha overall wet- i s now being vietrd illor a f l u t t e r standpoint, I,.,, looking fllor the &ace into tha apt-, tho hydlaulic a p t - ingat, % , , is marely a distprbsnca ami m y ba eonoidored 5.m.

Tha cbpracteriotia doteminant may also be axpressad,

4 - -.( A?,+ f (4S'f 43 +4 + Then tha capplote transfer function ralating rurfaaa load ami

mrfaae nation (with e ) is:

It w i l l now be shrnm that the quadrai;ic tern of equation (49) includes the only dyneaaical parameters considerd in the mlaal f lu t te r -08, where as the f i r s t tern, 4, 4+/4++ , cdpreaents the correction that must be added t o take into account the heretoion dirregarded chanrcteristics'of the hydnrulic-flutter s p r t a .

In oldiaary f lu t te r calculations it i s the practice to nae Theodorean~s differential equations of motion, The equation coq,nssing the .qrrillbrlum of the mcm~ents on a movable &ace is given by:

when oc = angle of attack

/B - marface angular deflection

X = vertical coordinate of ads of rotation of a i r ~ o h

j; = m m t of inert ia per unit length of ~u r i ace ( a h t hinge m e ) - s ta t ic mat of surface (in slugs-feet) per unit length (about hfnge f i e )

5 = torsional stiffness of eurface per uait lcmgth (about hinge -1

Mp = to ta l aezwlynamic manmt on sariace per unit length (about hfnge l ine)

IL = coordinate of a d s of rotation of airfoil (percent of a i r fo i l half chord length)

A = hlf dmrd length of a i r fo i l

c = coordinate of surface hfnge lfne (percent of a i r fo i l half cord length)

Fig. l4 U l s t M t a a &tr,wg.o~etrical p a m e t era of the girfoil-surfaoe eabinatimo

Of especid concern i s that portion of the total aelPdyrremic n e n t wbich i s related to ,9 and its derivatives alane. Designating this narmt oaapoor-t by yF ,

The above equation, obriowly, does not include the dynamical chsmcter- isticr of the hydraulic eystsm, but, rather, represeats a ourface inertla tied to a -ring, as illustrated belaw,

F1 Since MF s - 4

tp = 4 1' ( 4 include. a U flaxibil l ty I- the a! hydraulic cylinder to the effective mass

cmter of the surface)

A' = surface horn radius

d = surface span

then in meal terms, equation (51) becomes:

Caparison of equation (52) with (49) clearly sbua that the camplete apresrion for the hgdraulic-flutter sfst- is f a r more oarpl~x than the rimple situation shown i n Pig, l4.

The above ca~~ponant of the to ta l aerodyneaic =ant n l a t sd to and i ts derivatives a h e now replaces the orfginal incamplets tern

in equation (50); thenfore the differential equation of motion i r more precisely ~3q)ressed,

p diagraa~at$c r e p r e r m t a t h of the corrdcted capansnt i r ram . in Fig. 16.

A desirable f a a t a n of any well designed hydraulic aatrretor instal lat ion is t h a t the piston and mounting structure are a8 righi a8 porsible. Normally, then, the piston spring constant 18 large acupared to a l l other spring aonstants, including that of the hydraulia fluld. With t h i s important asmnnption, the quadratic (;*G sa 4 4 ++Js +S] becanes an appmxbate factor of the numerator and denaninator of the 4+ /A+ + portion of oquation ( 5 5 ) , reducing it to:

V a r y l i t t l e more can be done t o slnpUfy the above arprersion; actual values of the constants mast be substituted and the s3q,resrion nmuerically factored, Experience with typical Uorthmp-typo W r a u U c sptans, howover, bas shoun that the den-tor cubic has gn appnudYte quadratic factor, ( 4 s a + 4 s f 4 +I) j ; this im lies that the constant , is rn). large (1.e.. G i a very -7, uuch 18 the case fo r valve positions near neutral. Thus,

The term in the braces, t o rei terate, is the cornct ive t e a uhiah must be added t o (or mubtracted i n n ) the quadratic, f#= s a + 4 s +AG) , t o obtain the proper transfer function bet- the ae-c force, F , and the corresponding surface deflection, Xs . Thir corrective tern is of the form,

Frequancy Response of

Another prorequisite of a f ~ - p o w e n d bgdraulia r ena is teot it must be capable of resisting l a r ~ e hinge manar tm with relatively a m l l valve displaemmtr. It follars that the slop of the prosmum tr valve error cun. must be rather rteep, or that the value of the rpriry constant ( = &+/+ ) is hi . On the other M, the o i l spring eonstant 4 (= A?/*' rim n s t r i e t d by the eyUPder d s e tht can be installact a thin a i r fo i l reetim. SO, in gmenl , 4 will be greater than 4 ; thus

Aldo, since the off sofive damp- constant 4r i a very largo (or 6 very a a ~ ) cappared to tb+ ~ U e r nurse #, ,

Fig, 18 reflects tho above older8 of magnitude of the froquanalu inmlnd. In the regions marked A) and A2, the comctive t o n u contain eurnthIJy sero phase angle modification to be applied to the -la roetor

( M , s 2 + 4 s + 4 ) , and, hence, ray be eonslderd g ~ n spring corrections, The r m g h marked B ir a transitIan msqp batman 4, and dt ; its upper Wt is d e t e m h d px-harLQ by the value of tho

s e m gain tern, ce , whlch is Inherent in the spring dj . (Tho larger the value of At , the higher beaxnos the froqutmcy ; the range B temlnates Ughtly beyond that froquonq.) In the ruy C the correction factor is camposed of a l l three mecbanieal' elqants, i,e., marses and dampers as well as spring capblnations, The n-t pwk, 4" (4 +-.6 / ~ c ) f , fortunately occurs a t a verg Mgh f n ~ a m q , uld gonerally rill not affect the f lut ter analysis,

d f i c i m t l y low f ~ u m ~ i e s , ire,, approaching s tat ic or rtady-state load conditions, the aerodynamic loiPd transfer funotioa becanes a pure spring conatant fonned by the eerier l i shgo of the 8pmr , d the effective m d c 80Z'VO 4

The spr- 4 , is effective over the f u e m c y ~ g a A$ and its m1U Is derirable fmn q one of tho -tion755), (56), or (57).

Sinoe f lut ter i tself implies a dynamical condition, tho u J o r of interest l ies above the range A/ In fact, the tnsnsition ngioa B also includes fmquentios wwch are still well bolw those of the

nost camonly enccmntered f lut ter d e e . The range A' , t h e d o r e , i r the one most pertinent to the apalysis, The oxact t m f e r h c t l o n / cannot be obtained merely by adding a corrective s p A q tern t o

the quadmtlc ( ~ , s ' f ~ s * - * ( C ) ; harover nunorim1 =loll.- tions for typical systans have shown that the oqairalent spring ia tho dynamical range Az is essemtially the series -ring cambination that would be obtained i f the W m u l l c s y s t a were dared es a passim pot- work. Sn othor ~rords, a t sufficiently Mgh fmquson:ies $@ 8.1~00 action of the syatm is g raa tb auppresaed, axxi t h e effectivr in the v 4 is

or, ii the pinton s p w A, i n f ~ t e ,

M o d stat ic tent8 (ioeo, loading the surface and remuring i t r da- fleatione) vill not pmvlde a n a t i n f a c t o ~ nrann for o b t h h g j tho aborr spring conetento

To numnarise,the following concluniosu ngardbg Horthrpp typo b@raulic-flutter n p t a n may be made:

lo A Wrau l i c n p t comctian factor r~ r t bo added routerbllq t o the no& A c e quadratic ( w,s'+ 4 s t4) t o obtain the proper load trrranfer function,

2. Only the range AI of Figo l8 is of i n t e n s t in tho p r o b 1 since the regions A, and 8 ark wl l b l o w atrl tho mnge c i n fa r above the normaw encomterod f l u t t u fnquacioa.

3, I n the nmg. of interest, Aa , the hydraulic vtr mar k viawed ensantially an a pan npring, an f a r an tho abnaae of phase lag t a m in the correction factor i n coacemed.

The magnitude of this spring mmntant i n a p p r w d r a t e tht of the nerles caabinatian of (1) the spring co~pUq, the hydraulic ~ W e r and the surface, (2) the hydmul3c o i l wring, and (3) tho pinton rod nprln$ ( i i finite).

A - cylinder a m .

a - coordinate of axis of rotailon of airfoil .

% - orifice area

a< - angle of attack.

Be .- cylinder-piston damping coef f ic ia r t . 8; - input damping

4 - piston-st ructure damping coef f iciont . 4,- aynthatic damping coeff i c i m t

Bs - surface-structun damping coafficiart . b - half chord length of airfoil .

8, - valve-cylinder damping coefficient ,

p - surface angular deflection.

a - orif ice coefficient.

Cp - torsional stiffness of surface p r anit length (about hinge m e ) . - slope of the f h - v a l v e displacaart curve.

Cp - slope of the fh-pressure differential c u m .

C, - cylinder-piston coulomb frict ional forcer.

C, - valve-cyljnder coulanb frictional forces.

c - coordinate of d a c e hiag. l h e .

fi - valve d i 6 p h c ~ 0 n t f- nmtml.

E*- rteady atate valve displacaent (opomting point).

L - valve displacament perturbation.

f - aerodynamic load a t surface.

P - gravitation@ constant,

(Zen, Dim.)

R;$

rn-IT

la-*

no*

(zero Dim.)

(Zero Dk.)

$ - B Z ~ W ~ pfY88WC)*

%r - presstam on either side of piston. a gw- pertq&ed p r u m n on either side of p f r t a .

bp- proastam drop acmsa orifice

& = volr~n.tric flow fran valve or flaw into cyIlador.

a*= stead7 state mlu&ri!2 nln n ~ .

p - volumetric valve flaw perturbation*

p - fluid density.

S = Iaplaco operator,

& = s t a t i c manant per anit length of a i r fo i l (wlfh rwpect to 6ude of rotation of airfoil) .

9 = s ta t ic manant per rmit length of mrface, (with respect to eurface hinge line).

t = t h e .

v; = input vslocitJ.0

y = input v.locit~. perturbationo

w = specifia woight of fluid.

5.. = input velocity portul.bstion (velocity soume).

X,. = input displac.nent n l a t i n to atmcturo.

Xi = input diaplacenrnt perturbation.

& = cyllndor displacmmt relative to structuro.

X, = cylhder diaplacaaent perturbation.

XF= pl6toa ciisplacemmt relative to structure.

+ - piaton displacmuent perturbation.

X, - surface dlsplaccment relative to rtmotrur. (measured at cylinder horn).

D i r a r i a u

X, = ourface dirplacaont perturbation. L

X; valve d i r p l a c ~ t relative to rtructura. L

X,= valve displacaent perturbation.

Y = volrm..

8' = effective o i l voltme within cy~lnder.

p= 8 . a ~ skate 0f l VOl-8.

r/,I volrm. on either ride of pistono g

T = time conatant,

w = angular f nquency.

w, = natural a&hr f roquetlq.

J - -ping ratio

IJOTB: A l l dirplacaentr, velocities, acaeleration, mrses, s p r i q canstantp, etc., a n effectir. valuer rwrsand aoincidatt with or parrsUel to Ilae of aa t im of ~ I l n d e r .

(1) Feensy, To A., Vayermd Control8 Design Practiw a t llorthrop A i ~ ~ f t W S NO-+- Aircraft, Inc., 1949.

(2) Lin, So N., "A Method of Successive Appmxiaetione af Evaluating The Real and C a m p l r r Boots of Cubic and Higher Order EqrrPtianrn, Journal of M a t h t i c s and PhJrsics, Voltae XX Uo. 3, A u g u t 194l.

(3) Th.old0r8.n~ To, wGmneral Thwm of Aerodyna~Ic In8tabillty and the H e c b a n i ~ ~ of FlutteP, UACA TR 496, 1934.

(4) Meher, Do To, W A n Aualysis of l o r t h p Airerait Parerad PU@ ContmlsW, l o r t b p Aircraft, Into 19490

(5) Ino3Jsis of m e F-5 Autaaatic Pilot Applied to the Type P-89 Aircnrft and Control Systan - Contract AP 33(038)-6946, llerthrop Aircraft, Into, 19500

PARAMETERS FOR THE DESIGN OF HIGH- SPEED HYDRAULIC SERVOMOTORS

by HAROLD GOLD EDWARD W. OTTO

VICTOR L. RANSOM of

National Advisory Committee for Aeronautics Lewis Flight Propulsion Laboratory

Cleveland, 0.

INTRODUCTION

The servomotor dea l t Kith i n t h i s paper is a power amplifying, positioning device of the type used i n such applicatione as control valve positioners, f l i g h t controls and power steering devices. The hydraulic servomotor as a device has been known fo r approximately one hundred years. Its application t o high speed machinery, however, appears t o be re la t ive ly recent. There is consequently very l i t t l e published l i t e r a tu re on the dynamics of *is servomotor i n sp i t e of its long history. Nevertheless, when properly designed, t he hydraulic servomotor is part icular ly sui ted f o r high speed service because of t he extremely high force-mass r a t io s t h a t can be obtained and because the device inherently is heavily damped.

This paper is based on an experimental and analyt ical study of t he dynemics of the hydraulic servomotor recently completed by the authors at Lewis laboratory and soon t o be published by the NACA (reference 1) The r e su l t s of t h i s study have shown tha t although the response of t he servomotor i s essent ia l ly nonlinear and discontinuous, the response may be closely approximated with re la t ive ly simple l inear equations. The present paper presents data demonstrating the nonlinear, discontinuous nature of t he dynamic response of the servomotor and inqludes a deriva- t i on of t he baeic analyt ical expressions f o r describing the response. The derivations wfiieh are presented i n the "Analysis" section of this paper are essent ia l ly an abstract of the detailed derivations given i n reference 1.

The derived analyt ical expressions are summarized i n the form of charts. By means of these charts t he rat ional design of the servomotor t o meet specifications on e i ther the t ransient or t he frequency response character is t ics is made possible.

DEFINITIONS AFSD IKMT& ASSUMET'IONS

Straight l i ne servomotor. - The elements of t he s t ra ight l i n e hydraulic servomotor are shown schematically i n f i m r e 1. I n the nkutral position, the spool member of the p i l o t valve closes the paesages

HYDRAULIC SERVOMOTOR

SUPPLY PRESSURE

-7 OUTPUT SHAFT U

ROTARY HYDRAULIC SERVOMETER

PILOT VALVE e OUTPUT SHAFT INPUT SHAFT

-597

FIGURE 1

t o the piston. When the spool member is displaced from the neutral posi- t i o n by mvement of the input lever a t point (A), the flow of f l u i d through the p i l o t valve causes the piston t o move i n the direction which returns the spool t o the neutral position. It follows from the geometry of the linkage tha t f o r every position of the linkage point (A) there is a corresponding equilibrium position of the piston. The description of several other forme of p i l o t valving and feedback linkage is available i n the l i t e r a tu re .

Rotary servomotor. - The rotary servomotor i s also shown schematically i n f igure 1. Rotation of t he p i lo t valve with respect t o the output shaft opens a pressure passage t o one s ide of the vane and a drain passage t o the opposite s ide of the vane. The vane is thereby caused t o ro ta te i n the same direction as the p i l o t valve. I n the neutral position 'of the valve the passages t o e i ther s ide of t he vane are closed.

Initial assumptions. - The analysis which follows is developed with t h e following i n i t i a l assumptions t

1. The area of opening of the p i l o t valve varies l i aea r ly with t h e motion of t he valve.

2. A t f ixed input, t he r a t i o of p i l o t valve mvement'to piston mve- m n t is constant.

3. A t all positions of the p i l o t valve the i n l e t and discharge open- ings a re equal.

4. The supply and drain pressure are constant.

5. The comprese ib i~ ty and mass of the hydraulic f l u i d are negligible.

6. Structure and mechanical linkage are r ig id .

7. Mechanical f ract ion i s negligible.

'8. Fluid f r i c t ion losses i n motor passages are negligible.

9. Leakage is negugible.

Response t o a Step Input

Baeic dynamic considerations. - Under the condition of no load on the output shaft and negligible internal motor mass, the pressure drop across the piston wi l l be zero. The f l u i d flow through the cylinder is therefore

essentially unobstructed. The flow of f luid i s then proportional to the area of the valve opening and the flow coefficient . A t constant flow coefficient therefore the piston velocity i s directly proportional t o the valve opening. W i t h rectangular valve ports the open area of the valve is directly proportional to the error of the piston position. The velocity of the piston i s consequently proportional t o the error. I n the no load case therefore, the servomotor wi l l exhibit a traneient response that i s characterized by a linear f i r s t order system.

The pressure drop across the motor i s divided equally between the in le t and drain valve ports.

Based on a constant flow coefficient, the piston velocity may then be equated t o the flow through the valve ports by the , following relation:

Equation (1) may be written in the form

where

The analysis of the response of the se~omotor under an iner t ia load i s considerably more complex than i n the no load case. Under an iner t ia load the piston i s accelerated from zero velocity. There i s consequently an initial period i n the response during which the flow through the- valve ports is laminar, as a result of which the flow coef - f icient i s subject t o large variations. After the piaton hae reached the laaximum velocity in the transient, the momentum of the load may cause the flow of f lu id into the upstream side of the cylinder t o cavi- t a te . The variation of the pressure drop across the piston is therefore not describable by a continuous function and consequently a single differential equation cannot be written for the entire traneient.

In spite of the complex nature of the response there are basically only two phaees in the transient: the acceleration phase and the deceleration' phase. This conclusion, particularly with reference t o a continuous deceleration phaee without overshoot or oscil lat~oii , i s baaed on the assumption of rigid o i l and structure end zero leakage. Under these assuntptions the cylinder pressures during the deceleration phase are f i n i t e but may exceed physical limits.

Figure 2 shows an oscillographlc record of the response of a servomotor to a step input under a relatively heavy iner t ia load. The traces shown are: position responee, timing mark, downstream, cylinder pressure, and upstream cylinder pressure. The characteristic acceleration phase and dead heat deceleration phase are quite' clearly demonstrated. It will be noted that the downstream cylinder pressure exceeds the supply pressure in the deceleration phaee and a t the same time the upstream cylinder pressure i s driven to zero (cavitation occurs).

Linear equation for approximation of the acceleration phase of the responee. - It i s indicated by the measured reeqonee of hydrdl ic servomotors under inert ia loads that the acceleration phase may be approximated by a linear second order systein. The general form of a second order differential equation with constant coefficient may be written

A t m load the servomotor responds as a f i r s t brder system. Equa- t ion (4) should therefore reduce t o equation (1) for the inertialess case. meref ore :

A t the start of the transient ( t = + o), the upstream cylinder pressure I s equal t o the supply pressure and the domtream cylinder pressure ia kual t o the drain pressure . Hence when

Substituting these values i n equation (4)

m e differential equation that approximates the acceleration phaee i s then

Equation (5) may be def b e d i n terms of the no-load time constant T and the reciprocal of the damping ratio. This quantity i s here designated the "inertia index - E." The new term i s employed i n th i s paper becauee the term "bmging ratiow or other t e r n sometimes associated vi th the reciprocal of the damping ratio would have no meaning i n the type of transient associated vith the hydraulic servomotor.

Equation (5) expreseed in terme of the parameters T and E i s

Equating like coefficients i n equatione (5) and (6)

Substituting equation (3) in equation (7) the general expression for E i s obtained:

Linear equation for approximation of the deceleration phase. - I n the deceleration phase of the transient the pilot valve areas are reduced t o small values. Consequently the flow rate through the vdves i s affected to a far greater degree by the reducing value area than by variatione i n the pressure drop across the valve. For th i s reason the reeponee i n the deceleration phase does not deviate si&ficsatly from the no-load reeponee. Equation (2) may therefore be used to approdmate the deceleration phase.

Evaluation of coefficient for the rotary. servomotor. - By means of a parallel development expressions for T and E can be obtained for the rotary motor. These expreesione are given below.

l ication of equations. - The method of applying equations (2) the calculations of the traneient i s presented i n figure 3.

As ahown i n figure 3, the end of the acceleration phaee 1; defined by the inflection point of the solution of equation (6) for E > 1. For E < 1, the second order equation may be used t o approximate the entire t&mien t and therefore the inflection point need not be evaluated. Figure 4 i s a plot of r i se time against no-load t i m e conatant for a range of values of iner t ia index. The chart i s computed from the relations shown in figure 3. The r ise time i s defined as the time t o reach 90 percent of the f l n a l value.

erinrntal response. - Figure 5 shows the characteristic agreement bctwe?calcuiated and meataured responses (reported i n reference 1) i n a series of rune in vhidh the factors that determine the parameters T and E have been varied. Ae can be sten, t& calculated r e h n s e e have provided a close agpmdmation of the actual responses over a wide range of conditions. It may be of particular interest t o note that the effect of magnitude of the input step ars predicted by the approximating tiquatiom is evident in the measured responses.

Peak cylinder pressure during a transient. - In a transient under a heavy load, the downstream cylinder presaure may exceed the supply pressure. For th i s reason an evaluation of the peak cylinder pressure i s significant from a structural standpoint . It i s shown i n reference 1 that the rat io of the mnxrnnlm cylinder pressure t o the supply pressure i s a function solely of the inert ia index. The relation derived in reference 1 i s plotted i n figure 6. Also ihom in figure 6 are experimental values reported i n reference 1.

Response to a Sinu~oidal Input

Basic dynsmic considerations. - It has been shown that under an iner t ia load, the trans'ient response of the servomotor i s nonlinear, The basic character of the transient response has been shown t o vary with time in the transient. It can therefore be expected that the frequency response will also exhibit n o a n e a r characterXstics and that the basic character of the response will vary with the frequency of the input.

Law frequency amplitude attenuation and phase shiFt. - A t low fre- ",.s..ln4aa +'ha err..-.-., &La& --& ,̂ &I-^ ^o rL- ---- r-- --- ----a. - 1

CHART FOR DETERMINATION OF TlME TO REACH 9 0 PERCENT OF FINAL VALUE. HYDRAULIC SERVOMOTOR

WITH MECHANICAL FEED BACK

.NO LOAD TlME CONSTANT ( T I SEC

FIGURE 4

MEASURED AND CALCULATED TRANSIENT RESPONSES I

OF A HYDRAULIC SERVOMOTOR

65 - 0

EFFECT OF LWD INERTIA ,-GALCUTLATE! RESTONSE

FIGURE 5

The solution t o equation (11) , is

i B The term Ae is a vector quantity having an' amplitude A and

a phase angle fd. From equation (12)

High frequency amgUtude attenuation. - A t no-load the piston veloc- i t y is a t all times proportional t o the valve opening; therefore i n the response t o a sinusoidal input a t no-load the pi lot valve area is zero at the ends of the output travel (the velocity 'being zero). Under an iner t ia load the piston velocity is not l inearly proportional t o the pi lo t valve opening and hence in the response t o a sinusoidal input the valve area is not necessarily zero a t the ends of the output travel. If a t high frequencies, the response of the servomotor is assumed t o be essentially sinuaoidal, the maximum acceleration can be considered t o occur a t the L i m i t s of the output travel and hence when the piston veloc- i t y i e zero. Under the condition of negligible mass of the hydraulic fluid, the pressure difference across the piston a t any instant when the piston velocity is zero and the pi lo t valve area is greater than zero, is the pressure difference acrose the servomotor. Above some frequency the system may then be approximated by a Unear system wherein the pres- eure difference across the piston varies sinurroidally with an amplitude of (P,-P~) and with the frequency of the input. On the baeie of this app~oximetion the acceleration of the piston is

Integrating equation (15) and introducing the condition that x varies about zero and neglecting changes in sign

Dividing both sides of equation (16) by the output amplitude a t zero frequency, the equation relating the amplitude ra t io and the frequency l a :

The dimensionless quantity, iner t ia index, may be defined for the frequency response by replacing the magnitude of the step (s) with the term amplitude of the output sine wave a t zero frequency (s ') . Rewriting equation (8) and introducing the symbol (s') in place of (s) the expression fo r the ine r t i a index for the frequency response i s obtained:

: ~ q u a t i o ~ (3) and (18) substituted i n equation (17) yield the general expression for the amplitude attenuation

High frequency phase shif't. - The derivation of equation (19) does not provide relations by which the phase shl f t may be computed. The amplitude attenuation expressed by equation (19),- is , ho&er, the asymptotic relation of a linear second order system. The phase sh i f t i n the high frequency band may therefore be considered t o be described by a linear second order system. It can be shown further that equation (6)

Straight Une representation. - The straight l ine approximation of the frequency response relations i s presented i n figure 7. In the low frequency band the amplitude i s expressed by the aaumptotes of equa- t ion (13)

RATIO OF PEAK TRANSIENT CYLINDER PRESSURE T O SUPPLY PRESSdRE AS A FUNCTION OF INERTIA I N D E X

I HYDRAULIC SERVOMOTOR W l T H MECHANICAL FEEDBACK

PZ M A X - E~ eZ - - - Ps 14 77

WHERE K = f i

4' a / a - ANALYTICAL RELATION

a 0 MEASURED VALUES

= APPROXIMATIONS OF ASYMPTOTES

",., TO ACTUAL ATTENUATIONS 1

INERTIA INDEX ( € 1 v

F I G ~ E 6

SUMMARY OF LINEAR RELATIONS FOR M E FREQUENCY RESPONSE OF HYDRAULIC SERVO-

MOTORS WlTH MECHANICAL FEEDBACK fi

Lducr: , , DE,CAlXS PER DECADE

, , , - S T R A ~ H T LINE APPROXIMATIONS OF

- ,,:[ 1 ACTUAL PHASE SHIFT - e II[--j\l--66 DEG PER DECADE

DECADE

-- STRAIGHT LINE ROTARY SERVOMOTOR SERVOMOTOR

fpJ- T m A p J Z T , JZC:WCQ hlL1-L:)

2 x 7

-4 E'. R C R W

2' XTE

f3. X+,Z , , 4 c r ~ W -!w ( h IL:- LA)*

FIGURE 7 I

The high frequency attenuation is expressed by the re la t ion of equation (19) . The cross-over frequency i s defined by the intersection of the low and high frequency asymptotes.

Phase s h i f t i s represented by s t ra ight l ines on semilogarithmic coordinates i n figure 7 . The slope of the low frequency l i n e i s defined by the slope of the f i r s t order system (equation (14)) a t jd = 45O. The slope of the high frequency l ine i s defined by the slope of the second order system (equation (20)) a t $ = 90'. The low frequency phase s h i f t l i n e passes through # = 45' a t the low frequency band break frequency ( fl) . The high frequency phase shift l i ne i s oriented by the cross-over f r e - quency. I n figure 7 the cross-over frequency i s shown t o occur a f t e r the low frequency phase s h i f t l i n e has reached the 90° l i m i t . The or i - entation of the high frequency phase sh i f t l ine fo r other re la t ive locations of the cross-over frequency i s shown i n connection with the experimental responses.

Experimental responses. - Figure 8 shows the experimentally and analyt ical ly determined effect on the frequency response of the hydraulic servomotor of the parameters: load ine r t i a and input amplitude.

Figure 8(a) shows the ef fec t of load ine r t i a on the amplitude attenuation and on the phase s h i f t . An increase i n load i n e r t i a resul t s i n a reduction i n the frequency a t which the attenuation becomes rapid. I n the analytical expression developed i n t h i s paper (summarized i n f i g . 7) th is e f fec t is evident i n the increased value of E' with increasing load i n e r t i a and the consequent reduction i n the values of f 2 and f3.

Figure 8(b) shows the ef fec t of input amplitude on the frequency response. The increase i n input amplitude i s seen t o have an effect sim- i l a r t o tha t of increasing load iner t ia . This effect i s made evident i n the analysis by equation (17).

I n both amplitude and phase s h i f t the agreement between the measured responses and the analyt ical s t ra ight l i n e approximations i s i n general well within the experimental accuracy. The slopes of the attenuation and phase data clearly demonstrate the f i r s t order character is t ics of the response in t he low frequency band the the second-order character is t ics of . the response i n the high frequency band. The t rans i t ion from f i r s t t o second order character is t ics a t the calculated break frequency i s quite pronojmced.

APPLICATION TO DESIGlV

Design Relations

From equatiol23 (3) and (8) the following expressions f o r the piston area and t he product of the feedback r a t i o and port xidth of a s traight l ine servomotor can be derived:

Equations (21) and (22) are written i n terms of t ransient response param- e t e r s but apply t o frequency redponse parameters by replacing S and E with S t and El, respectively. The corresponding equations f o r the varioue s izes of a rotary servomotor i n terms of the various design parameters may be obtained from equations (9) and (10) i n a similar menner .

Equation (21) expresses the re la t ion between the motor i n e r t i a ( ro- portional t o 9) and the various response parameters. Equation (22 expresses the re la t ion between the input i n e r t i a (proportional t o RW and the various response parameters. The motor ine r t i a and t he input

I i n e r t i a a re signif icant factors i n the design of high speed servomotors . The following paragraphs wi l l diecues methods f o r aiding i n the selection of optimum designs t o meet specifications on e i ther t he t ransient o r frequency response character is t ics .

Determination of Design Parameters

Load mass. - The value of M i n the design equations includes the mass of the output par t of the servomotor as well as the load mass. I n high-speed applications the motor mass may be a significant percentage of the t o t a l mass. For this reason the estimated mass of the output par t of t he motor should be added t o the known load mass. It is obvious from the character of t he response of hydraulic servomotors of this type tha t t he , responee~,characteristics f o r a given s ize are always improved by reductions i n the load mass.

Pressure d i f fe rent ia l across servomotor. - I n general the s ize of a servomotor fo r a given response i s reduced by an increase i n pressure d i f - f e ren t i a l . The reduction i n weight, however, i s modified by the need f o r larger sections t o withstand the increased pressure. The determination of the optimum pressure' i s beyond the scope of t h i s paper. The no-load time constant varies ' inversely as the square root of the pressure d i f - ferential'whereaa the i n e r t i a index is independent of the pressure d i f - f e rent ia l . Therefore i n a given servomotor the break frequencies. (f and f3) a re proportional t o the s q w e root of the pressure d i f fe rent ia l , and the r i s e time i s approximately inversely proportional t o the square root of t he pressure d i f fe rent ia l .

Magpltude of the desired output. - Because the character of the response is determined by the magnitude of the desired output, a value of the output magnitude must be selected fo r design purposes. I n general this information is obtained from the process which the servomotor i s varying and consists of an estimation of the maximum percent change i n the process tht is required t o occur i n a given traneient or s ine wave osci l la t ion. This variat ion should then be t ranslated in to the output stroke required of t he servomotor t o produce such a change. The t o t a l stroke of the servomotor wi l l usually be larger than the design value of S o r S t and corresponds t o . t h e steady s t a t e range through which the process i s carried. I f a s tep change or s ine wave osc i l la t ion larger than S o r S t is introduced the resul t ing i n e r t i a index wi l l be larger than tha t assumed i n the design with the consequent poss ib i l i ty of damage from high peak cylinder pressures. Protection against such damage can be obtained by r e s t r i c t ing the length of the p i l o t valve lands.

Selection of no-load time constant and i n e r t i a index f o r frequency response requirements. - I f the frequency response requirements of, the servomotor are known they i n general w i l l take the form of the f i r s t break frequency ( f l) and the &oesover frequency (f3). The equations fo r T and E' i n t e r m of these two frequencies a re as follows:

As shown i n the analysis the response of a servomotor of t h i s type i s characterized i n both attenuation and phase s h i f t by a f i r s t order re la - t i on up t o the crossover frequency. The phase s h i f t i n t h i s frequency band is therefore limited t o a maximum value of 90'. The crossover f re - quency f 3 can therefore be selected on the basis of suff icient amplitude attenuation at 90' phase s h i f t t o insure s t a b i l i t y i n the control loop. With f l and f g defined, the dimensions of the servomotor are estab- l ished. Substituting equations (23) and (24) i n equation (21) the fo l - lowing re la t ion i s obtained:

2 4MS1p flf3 (251

From equation (25) it can be seen tha t at a fixed value of f l an increased margin f o r fg can only be obtained by a proportionate increase i n piston area.

Selection of no-load time constant and i n e r t i a index fo r t ransient response requirements. - Transient response requirements can be expressed i n t e r n of the time t o reach 90 percent 'of the f i n a l value. The

variation of th i s r i se time with T and E was presented i n figure 4. It can be seen fmm figure 4 that a given value of r i se time can be obtained a t various combinations of T and E. The relation presented graphically i n figure 4 may be exptessed approximately by the following equation :

Equation (26) combined with equation (21) yields a general relation for the variation of piston area with iner t ia index. This relation i s :

Equation (26) combined with equation (22) yields a general relation for the variation of the product of the feedback ra t io and the port width with iner t ia index as follows:

2 3 The terms (i + $1 and BG + i n the above equations are proportional

to Ap and FW, respectively. Thus the numerical values of these terms plotted as a function of E will indicate the variation of the size of the servomotor with iner t ia index. The resulting plot i s showq i n figure 9. Ae a further aid i n the choice of a value of E the maximum cylinder pressure which i s a function of E i s also shown.

Figure 9 indicates .that values of E above 6 do not materially reduce the size oaf the servomotor and result i n large values of msximum cylinder pressure. Values of E below 1 result i n large servomotors and i n general would be used only i f it i s desirable to make the response substantially f i r s t order. In t h i s respect the above curves indicate that it would be very diff icult to reduce E t o values close t o zero because the motor size and thus the mass of i t s parts become large, which w i l l tend t o increase E. Values of E i n the range f r o m 2 .to 4 should result i n a servomotor that i s very eatiefactoryboth from a size standpoint @ , from a response standpoint.

EFFECT OF LOAD INERTIA ON THE FREQUENCY RESPONSE OF A HYDRAULIC SERVOMOTOR WlTH MECHANICAL FEEDBACK

:ALCULATED STRAIGHT LINE APPROXIMATIONS

FREQUENCY-CYCLES PER SECOND

FIGURE 8

EFFECT OF AMPLITUDE ON THE FREQUENCY RESPONSE OFA HYDRAULlC SERVOMOTOR WlTH MECHANICAL FEEDBACK

CALCULATED STRAIGHT UNE APPROXIMATIONS

FREQUENCY- CYCLES PER SECOND

FIGURE 9

CHART ILLUSTRATING EFFECT OF INERTIA INDEX E ON SERVOMOTOR SIZE

FIGURE 10

CONCLUDING l3mARKs

The discussion has shown that the response characteristics of hydraulic servomotors with mechanical feedback may be represented by linear equations. These equations are defined in terms of the size of the servomotor and the load conditions. From these relations, equations which define the dimensions of the servomotor i n terms of the known load, and desired response characteristics can be obtained. BY means of these relations the rational design of the servomotor to meet specifications on either the transient or frequency response characteristics is made possible.

1. W A Report. Dynamics of Mechanical Feedback-Type .Hydraulic Servo- motor under Inertia Loads, by Harold Gold; Edward W. Otto, and Victor L. Ransom (soon to be published) .

APPENDIX - bsYMmLS

The following symbols are used i n t h i s analysis:

r a t io of output amplitude a t a given frequency t o the output amplitude a t zero frequency

piston area, sq in.

constant

constant

ine r t i a index (transient response)

ine r t i a index (frequency response)

low frequency band break frequency, cycles/sec

high frequency band break frequency, cycles/sec

cross-over frequency, cycle~/sec

width of vane, rotary servomotor, in .

polar moment of iner t ia , lb-in. sec2/rad

inner radius, rotary servomotor vane, in.

outer radius, rotary servomotor vane, in .

lomi mass, l b sec2/in.

upstream cylinder pressure, lb/sq in . abs

downstream cylinder pressure, lb/sq in . abs

drain pressure, lb/sq in . abs

supply pressure, lb/sq in . abs

r a t i o of valve t ravel t o piston travel a t fixed input

r a t io of valve t ravel t o vane shaft rotation a t fixed input, i n . / r d

magnitude of step (measured a t output), in .

amplitude of output sine wave a t zero frequency, in .

pa lo=$ $8 uoy$eyQd moy pamwam $ndqno JO uoy$yeOCI moa~[~$uw$euy

pw;r 'Xmanbazj oaan $8 a~a arqe $nd$no JO apn$-mBm

pw;r '(qndqno $8 pamwam) da$e JO. apn$m

*ny 'pq $8 uoy$yeod may pammam $nd$no JO uoy$yeod moaae$m$euy

3ae '$uayew;r$ JO may amy$

IMPROVEMENT OF POWER SURFACE CONTROL SYSTEMS BY STRUCTURAL DEFLECTION COMPENSATION

JAMES J. RAHN EVERETT W. KANGAS

Boeing Aircraft Corporation Seattle, Washington

This paper i s based on original work done by Mr. Howard Carson of the Boeing Airplane Company. The authors gratefully aoknowledg the cooperation and assistance of their Engineering colleagues a t Boeing Airplane Company in preparation of t h i s paper.

IMPROVEMENT OF POWER SURFACE CONTROL SYSTEMS

BY STRUCTURAL DEFLECTION COMPENSATION

Power controls for operation of aircraft control surfaces have become a necessity as a result of increased airplane sise, speed, range and control requirements. The direct feedback linkage type of powered surface controls presents the problem of s t ab i l i ty versus respome. Thie i s especially true when a large surface inert ia i e coupled v i th a relatively l a w structural spring rate. Although these systems nrny be eatisfactory fo r manual operation, they msy prove marginal or u ~ a t i s f a c t o r y when used with automatic f l i gh t control equipment.

A moans for improving the performance of a mechanical-Wdraulic control sy r tm hoving high surface iner t ia aad low structural spring rate8 has been dweloped and has proven quite ~uccessful on an airplane installation. In this discuseion we w i l l f i r s t cover a basic type powered surface control system; Them it w i l l be s h m how a rtructural deflection feedback linkage can be added t o the cystom to imprwe performance.

A BASIC TTPE PCMER OPERATED CONTROL mTEn

A powered surface control system operating a high iner t ia surface which had a low reaction spring ra te m e ~ y be limited i n performance. To retain s t ab i l i ty of the system the gain must be kept low, resulting i n slow response which m y make the system Inadequate fo r 8atisfactoi.g airplane control. T h i s problem was encountered i n the dwelopment of a rudder control system which is described belaw and s h m I n figure 1. operation of the p i l o t ~ s pedals rotatee the torque tube X i by means of cables. A differential linkage i e l 0 ~ 8 t e d a t the upper en3 of the torque tube. Since the surface i e i n i t i a l l y stationary, the movement of the torque tube causes movement of the error l ink X, which i s connected t o the W r a u l i c control valve. Displacement of the control valve admits pressure t o one side of the piston ecld opens the other side t o return. The resulting pressure differential supplies the force t o mom the ndder against inert ia , fr ict ion, damping and a i r load. The follou-up or feedbncK l ink connected to the rudder and t o the differential 3 then moves the ifferential to reduce the error t o a minimum and closes the tgdraulic control valve. When the CIVW is a dniru, tb. valve i e i n neutral and the f luid trapped on both side8 of the pieton holds the rudder i n one position against variable loads.

This eystem as f i r s t instal led i n the airplane, fignre 2, prwed t o be unstable. Any disturbance t o the control surface or to the pi lo t input resulted i n a continuous oscillation of the eystem. To overcome this ins tabi l i ty the system was slowed d m by reducing control valve gain. The net result wae a deterioration i n performance and excessive phase lag. The combined lag of the

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power operated syetem and tha t of the airplane made p i l o t opera- t i on d i f f i c u l t and operation with the automatic p i l o t unsatie- factoxy.

An analyeis of the problem indicated tha t the d i f f i cu l ty could be at t r ibuted primarily to the reaction epring ra tes of the sgsten. The structure of the reaction vetem had been deaignqd f o r strength in accordance wlth anticipated loads and not f o r r ig id i ty and thus accounted f o r a large portion of the elast ici ty. The compressi- b i l i t y of the o i l along with m a n s i o n of the power cylinder, I ~ ~ I r a u l i c tubing ard other components a l so contributed to the low overall spring ra te of the qstem. Bending of a m and link- age8 i n the feedback system a d entrained a i r in hydraulic o i l were recognised as other factors involved.

1t was also found tha t the original ralw, although providing required gain a t higher amplitudes, caused too high gain a t low amplitudes resulting in instabi l i ty . Other probleuw included backlaeh uxl play in the system which was attributed primarily to bearings and the movenent of the power cylillder piston "0" r ing In its Nstardardm groove.

Various changes t o the syetem were made in an e f fo r t to improve performance. These included the design of a control valve whose gain varied a s the square of valve displacement, use of close tolerance bearings a d bolte throughout, and use of more r ig id feedback a d control linkages. The reaction structure was also etrengthened locally, but it was found that a large portion of the f i n contributed to the spring rate.

Althaugh some improvement was real i ted from this effort , it w a s apparent t M t the performance was still l e s s than desired. Fig- ura 3 shows the frequency response curves of amplitude r a t i o a d phaee of the sgstem with these inprovements.

The natural frequency of the airplane upon which this control system was being used is shown by the dashed ver t ica l line on f igure 3. .It can be eeen tha t the pawered surface control eye- tam p h s e l a g wae 70° for 1 1/4 degree eurface amplitude. For smaller surface amplitude8 the phase l a g increased rapidly. I n this rmga of unplitndes the combined phaee l ag of the control syetsm and tht of the airplane approached a value so high tha t m f f l c i e n t phaee lead could not be realised from m automatic p i l o t t o keep the closed loop sgstem stable. To be on the safe s ide the powered control system should ham l e s s than kS degrees phase lag, f o r q amplitude a t which the rudder is e f fec t iw , a t a frequency 5 times t h a t of the airplane natural frequency.

AMPLITUDE RATIO- DB I 0

I 0 I

cn 0 cn 0

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~ d d i t i o n a l means were t r i e d i n laboratory t e s t s t o improve performance. Damping i n the feedback loop was tr ied, although be t t e r results were obtained, the ins ta l la t ion complications did not warrant its use. The use of a double om rlng piston a l so helped t o reduce phase l a g by reducing backlash of the system. Although the improvement did not reach the magnitude desired, the double ROW r ing piston was instal led since bacldash u i l l deterlorate aw system.

The main problem, t h a t of low spring ra tes of the system, still remained, and prevented necessary increase in system gain to obtain be t t e r response. T h i s problem can be i l lus t ra ted by referring t o a basic type power operated control system using a d i r ec t feedback linkage as shown schematically in figure 4. It can be seen t h a t

external disturbance t o the control surface o r mass results in deflection of the reaction structure. I f the input system is held fixed while the reaction structure i s deflected, the followup linkage opens the control valve i n a direction to oppose this disturbance. ~f the system has high gain, the control 'valve may admit enough energy t o cruse the system t o over correct the surface position. Due t o the ine r t i a of the surface the over correction causes a deflection of the structure I n the opposite direct ion which i n turn again opens the valve and again excess energy is admlttecl a d the cycle is repeated. I f the energy admltted per cycle is equal to o r greater than tha t extracted by aerodynamic damping and f r i c - t ion of the system, the surface w i l l continue to oscillate. The amount of energy t h a t is added t o the gystem is p r inud ly detelc mined by the flow character is t ics of the mstering v a h e and mry be kept smal l if the flow per uni t deflection i s emall. Stab i l i ty m e y , therefore, be achieved by ube of a law gain valve. T h i s improvement in stabi l%ty of the system is obtained a t the expense of speed of response.

A s very l i t t l e can be done about the hydraulic spring r a t e and a s the structure s t i f fness can be increased only t o a limited degree, it was evident tha t some other means should be coxwidered t o compensate fo r all of the system spring rates.

ADDITION OF STRUCTURAL COMPE2SATION TO THE BASIC TYPE SISTBl

High performance of a servomechanism cannot be obtained if lack of s t a b i l i t y l ind t s the allowable gain. Therefore, the s t ab i l i t y of the system is the gwerning factor. Improvement can be obtained by increasing e i ther spring r a t e or damping or by reducing the mass. For this system the mass could not be reduced and adding damping to the output was not considered pract ical o r desirable, leaving the reaction spring r a t e a s the factor t o be considered in order t o Improve s tabi l i ty .

The overa l l r eao t ien rpring r a t e i r due t o a r e r i e r of rpringr. In tho sohematio. f i g u r e 4, they were grouped i n t o 2 min rpringr, K8 r t r u o t u r a l rpring r a t e and Kh hpdraulio and other spr ing rater. The f i r r t problem i r t o f ind a r s n r f o r G a r u r i n g tho rpr ing deflmotion thon seeond t o f ind a ma- of ur ing t h i s rgr ing def leo t i en t o rtabiliw the rymtem.

The r t r u o t u r a l de f leo t ion i r a oonvmiently r a r u r a b l e quanti ty* It o m be ured t o represent the - t i r e rpr ing ryrtem d e f b o t i m if t h e propor r a t i o i r applied. H w t h i r d e f l e o t i m i r ured wi l l be 8h.m analyt ioal ly . However, in order t o provide t h e baris f o r the a n a l y t i o a l treatment of t h e subjeot, t h e rystem and i t 8 spera t i aa wi l l be explained first. R o o f f o r t h i r explanation wi l l then be given.

In t h i r r y r t e r a mohanieal linkage is used t e modify t h e ryrtem e r r o r i n aooordanoe r i t h t h e de f leo t i en of t h e reaot ien r t ruoture . Tha b a r i o system of f igure 4 rlth the add i t ion of rtru0tur.l d e f l ~ ~ t i o n oempensatirn i r rhcnn i n f i g u r e 5. AU ex te rna l dirturban00 t o the oontrol surfaoe o r maor rill def leo t t h e reao t ien r t ruoture* The follarr-up linkage rlll m o l s the oontrol valve t o oppore th. disturbanoe i n the same manner a8 i l l u r t r a t a d f o r t h e unoampenrated syrtem. The ooqtonratian linkage, horswp new a l r e add8 t e t h e valve rignal. The di reot ion of t h e valve movemnt due t o t h e oompenratiag l inkage i n opporite t o tha valve movement introduoed through the fo l lm-up linkage. By t h e proper urangsmsnt of t h e r a t i o between the reaot ien oompenrating linkage and t h e fol ls r -up linkage t h e d i r e o t i e n and m m i t u d e ef oontrol v a l m rovemont 01~) be varied. Tho mwemsnt of the oontrol valw a r a r e r u l t of' an ex te rna l dirturbanoe oan be aade t o reduoe tb amount of energy added t o the syltem or t o ex t rao t energy From the ryetom. For a r p o i a l linkage arrangemnt, energy rlll be ne i the r added to or extraoted from t h e ryrtem. The added linkage oarr be m d e t o r t a b i l i r e the ryrtem regard le r r of the m p i t u d e , d l reot ion or o r i g i n sf the d i r tu rb ing foroe. Thir pormitr m inoreare in system gain t e obtain improved p o r f o ~ o e .

An m a l y r i r of t h e ryrtem rhuur t h a t t h e ryrtem a t a b i l i t y i r depend- e n t on a derived term whioh rlll be referred t e a8 t h e 'Coqtonratien Faotor ."

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I n general, the value of Ks and Kf remain constant for a given system. The value of c a d d are usually defined by the geametzy of the mechanical input system and cannot be changed. By combining these tema the Compensation Factor reduces t o a cone tu~t times the reaction compensation linkage ratio.

To obtain a stable system the value of Cf must be equal t o ar greater than 1. Actually the system becomes less unstable a d inrprovement is obtained even for values of Cf l e ss than ond approaching 1.

The above derived term for the Compensation Factor and the c r i t e r ia for s tabi l i ty can be arrived a t mathematically as is e h m by the following analysis. The symbol deflnitioars are included on page 12 and, refer t o figure 5 .

The differential and algebraic equations fo r %he sgstsm are written as follaws:

Summation of forces on the output member

Sumfnation of forces on the reaction spstm

From the geametq of the linkage system

From the equation of continuity of hydraulic flew

Yv K.4 - A, ( x , + x ' h )

Substituting equation (3) into equation (4)

With theae substitutions into equation (7) the dl f femntia l equation8 for the systaa becorn,

Tho solution of these equrtions provldes the transfer -tion of the syr tm i n operator fonn

An examhatian of this characteristic equation by mane of Routb crit8rion, and by assuming tint aerodJmrwic damping is saw although it t.ds t o etabflise th syrtenn, it can be sham thnt the s tabi l i ty of th system i a depedent upon the 'Chponhation FautoP. The factor i a defined as

fo r the eystem being analysed. For the systcrar to be stable the value of Cf muet be greater than 1, Neutral s tabi l i ty is obtained when Cf equal 1,

By proper selection of linkage ratio8 the va lw of Cf can, of course, be varied. For the epecial case of Cf equal to 1 t b m wlll be sero deflection of control valve sl ide mlativm t o the s l e m for any dieturbance of the eystem. I f Cf is leer thn 1, the control mlva s l ide movement vill be in the direction t o oppore the oxt .mrl dieturbance, adding energy t o the system ond tending t o destabilise the - a h . This approaches the uncompensated - t a n M c h hrs Cf e q d to tern. A linkage ra t io resulting in Cf baing greator thn 1 rill cause the control valve to m o n in a direction not opposing the disturbance. This ixxlicater a 1088 of onrrgy fian the ayetom and

will result i n a damping effect on the power control 6 7 8 b .

Obviously, where the r tabi l l ty of the 6yrtQl i r of importame, oaly values of the capenut ion factor Cf equal t o or gna tor than 1 are of intereat.

To i l luotra te the .application of the Colap~lsation Faator to ob- a linkage ra t io for a stable rystem having Cf equal b 1, arauas tho follouing valwr:

Tbon t i ca l l y u tho ra t io of b t o a bocorur largor, tb s y r t r damping becomer gmakr , md u tho ratio get8 W e r it k d r t o dertabilise the syrtem.

Tho unc0mpamat.d sgrtom thmomtically i r armmod t o bd i n u n r s i b l e . Reverribillty here i r referred t o a r dirp&comnt of the m t p t d e r d bear8 no relatiomhip to tho proportional p i lo t fee l of control marface lod conaaonly refomad t o in boort ryrtans. Act- it is rrverrible a8 tho valve deadspot .ad leakage -0% be reducod to sero, backlarh emmot be ontirely olblnakd, .ad thom will alwsgrr be a f i n i k e&rt ic iQ In the sgr tm linkage. The companeated s y r t a i r tbore t ica l ly rwaruible. The degree of revarribil i ty d e r r t a t i c c d i t i o n r wi l l be equal t o Cf timer tho r m r r i t d l i t y of th. roaction sy&sm. 'For imtance,

C equal to 1 the m e m i b i l i t y of tho s y r b k i n g I d l r cu r rd r approdnrtely .9 degne for . m u h u m load, Under n o d conditions with tho surface a t o r now rtrermlill. porition, the rurface lord8 will actually tm a null porcent of the .urilmma load a d the rsverribil i ty w i l l t m proportionally mallor.

Although only one type of power oontrol syrtem has been dieoussed here, it should be pointed out t h a t t h e theory of s t r uc tu r a l d e f l e o t i m oompensation appl ies equally -11 t o ans basio poe i t ion ing sys tom.

A n a l p i s of the system has a h m tha t t h e s t a b i l i t y of the syrtem i s primarily dependent upon linkage r a t i o s . An eva lua t im of a system f o r a desired oompensation faotor rill provide an approxiap.ta value fo r t he required linkage ra t io . In an aotual i n s t a l l a t i on , it may be d i f f i o u l t t o aocurately determine t h e spr ing r a t e s of t h e system, and t e s t s mry be required t o obtain the bes t liPkopp r a t i o as governed by desired symtem perf onmnoe. This prooedure was used i n a r r i v ing a t the optimum linkage r a t i o f o r t he i n s t a l l a - t i o n disoussed i n t h i s paper.

A d i a g r w t i o sketoh of the basio pwer control system with tho addi t ion of t he oanpensated linkage i e ahom i n Figure 6 . In tb aotua l i n r t a l l a t i o n a s rhom in f igure 7 it was neoessary t o add f i v e l inks t o t h e or iginal eystem. The nwnber of l i nks required i s p r i l a ~ r i l y omt ro l l ed by the physioal location of oaapmenta i n t h e system. The a p p l i c a t i ~ n of t h i s type linkage in to a.sy8t.m already designed may be mere d i f f i c u l t than fo r a new system whioh oan be designed t o inolude the simplest and most e f feo t ive arrangemnt.

The improvement i n s t+bi l i ty of t h i s system r i t h t h s addi t ion of tho ompensating linkage permitted use of a higher gain valve with a near ly l inear flow versus diaplaoement oharaoterist io. This resul ted .in a higher performance system f o r a l l amplituder. The value of the canpensation faotor f o r t h i s system as i n s t a l l ed in t h e airplane was approdmately 1.1.

The degree of i m p r m m n t of the ac tua l a i rplane i n s t a l l a t i on i s s h m i n f igure 8. This i s a representation of the power oontrol syatem performance'. It can be seen t ha t t h e phase o h r a o t e r i r t i o r of t he r p t e m have been improved oonsiderab2y w a r t h e uuo0mpenaat.d syrtem as shown i n f igure 3. For b e t t e r oomparison, f igures 3 m d 8 a r e both shom in f igure 9. I t rhould be noted t h a t t h e amplitude a t whioh the oolapensated system was t a r t ed 1.8 only fl /L degree a s oompand t e tl-l/LO for the m o r p l u a t e d ry.tem. This t o ahow e m more t h e degree of i m p r m u n t whish m a obtained, a8 the r e a p s e of a system p n e m l l y beeomor poorer r i t h a deorease i n amplitude.

'9.

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In t h e range ai a i rp lane n a t u r a l frequenoy, t h e p h r e l ag of t h o p m r e d rurfaoe oontrol ryrtem h r been nduoed t o 20'. m n though tb ai rp lane phare l a g (rudder ncnenmnt t o ow) i r a p p r o r i r k l y 90' in t h i r Srequmoy range, tb t o t a l k g i r ruduoed t o a value ruoh t h a t t h e moer ra ry phare lead om be rea l i red in m automatie p i l o t in 0rd.r to athim ratimfactoq overrtl porforrurrre.

The f i n a l m l u a t i e n of t h e ryrtem, of oourre, depend8 on perform- m o e on a f l i g h t t o r t a i r p l u m . The ryrtem whioh h r been di rourred here proved suooesrful i n el iminating steady r t a k o r o i l l a t i a n r and improving maneumrabil i ty and p e r f o m o e f o r automat10 p i l c t operation.

1. The per fo rnnoa of a mohanioal-hydraulio rervo ryrtem i r l i a i k d by t h e reaot ien rpring rate.

2. This l imi ta t ion o m be emroom by u t i l i t i n g r t r u o t u r a l d e f l e o t i on feedbaok*

3. The r t r u o t u r a l d e f l e o t i s n feedbaok linkage o m be d e s i p u d t o aompensate f o r a11 d e f b o t i o n s of the r y r t m .

4. Capensat ion r e r u l t r in a r e r e r r i b b ryrtem proportional t e e l a r t i o i t y of maot ion struoture.

5. Higher responre of the ryrtem can be r e a l i r e d with r t r u o t u r a l def leot ian feedbaok r e r u l t i n g in isprored parform- .no..

A SURVEY OF SUGGESTED MATHEMATICAL METHODS

FOR THE STUDY OF HUMAN PIWT'S RESPONSES

by EZRA S. KRENDEL The Franklin Institute

Laboratories for Research and Development Philadelphia, Pennsylvania

During the l a s t decade considerable e f fo r t has been directed by several laboratories, both i n t h i s country and in England, toward the solution of the problem of ra t ional ly characterising the human operator of various type8 of control equipment. The goal of t h i s research has been the determination of what might be called humen "transfer functions," where the term transfer function is used in a loose 8ense. In general, the work has dea l t with problems ar i s ing i n gun direct ing situetione. Although re la t ive ly l i t t l e e f fo r t has been concetned with the study of the guidance problems ar i s ing i n the control of a i r c ra f t , the display and aontrol problems for sighting system and a i r c r a f t have suf f ic ien t simi- l a r i t y t o make the suggested methods of study applicable to both problem. The purpose of th is paper is t o s t a t e some of the p s t suggestions f o r appropriate ma thematical models to describe human operator performance, and t o present some of the current thinking a t The rh-anklin Ins t i t u t e on the problem.

The problem under discussion ar i ses from the control s i tua t ion wherein an operator is attempting t o or ient a device so tha t a visual ly perceived error may be minimized. Figure one represents t h i s s i tua t ion in a simple tracking task which was Fnvestigated a t The Franklin Ins t i tu te . The specification of the t ransfer functions in th is f igure is only for explanatory effect . The humen element in th is control system consiets of the eye, which is the primary error sensing mechanism; the nerves, which a re the data transmission system; and the arm muscles which a re the motor system. It is, however, necessary t o specify more about the s i tuat ion. Since the operator is largely non-linear, the stimulus must be careful ly specified. The visual s ignal can be characterized by the degree to which

- the operator is able t o predikt the stimulus. Close to perfect prediction can be achieved e i ther by having the future behavior of the s ignal disclosed t o the operator over a length of time greater than h is reaction tlme plus - movement time, o r by conveying very l i t t l e infomation in the visual stimulus by presenting the useful past history of a s ignal the future of which is determined simply from its past. ' Ekamples of the f i r s t type of eas i ly anticipated s ignal a r i s e in the driving of an automobile; whereas an exemple of the second type of predictable s ignal would be the tracking of a siapple s ine wave. The display which a gunner sees in his s ights or which a p i l o t sees when get t ing on ta rge t d i f fe rs from the foregoing in the t the possible

Figure 1.

anticipation of the ta rge t ' s motion is limited by the narrow f i e l d of view as well as by the conditions generating the motion, such a s evasive maneuvers and unforeseen random dls turbances due t o gust6 . It can be seen f'rom the foregoing examples tha t any study of human responses m u s t careful ly consider the nature of the visual Input because of thia anticipation character is t ic of the operator.

Another well-Imom character is t ic of the human operator is the a b i l i t y to adjust t o and compeneate f o r specif ic tracking problems and spec i f ic equipment. For example the operator is able to adjust h i s response to take into account f r ic t ion , different control r a t io s and other aspects of 'the equipment. Similarly the operator is able t o adjust h is response with regard t o instructions such a s "track accurately,""track rapidlyn o r in the case of a i r c r a f t f l i g h t to m e i n t a i n a smooth course through a ser ies of gusts instead of attempting t o compensate for each one. In addition the operator's behavior i s capable of changing as a function of bxperience and training in any par- t i cu la r tracking task.

The foregoing characteristice point up the d i f f i cu l t i e s in attempting to ar r ive a t a l inear , time i n v a r l n t , description of human operator behavior. I n addition ' th i s non-linear nature of the human operator makes any character-

imation of the humrn mom or l u r p r t i o u l r r t o the oiroume tama under which the mwsuromentr were mde. Thum, If a t r a m f o r or, mom p p o r l y , desoript iw funotion oan bo determined fo r the. operator, it aan only bo epeolfled f o r a given olasr of input sigpala, f o r a given pieoe of appl- rat-, a d fo r givon Inetruotiona and degree of praotioe. I n additlimp thb function w i l l exhibit variations ar is ing from the Individual di i fbr- encee in the sampled population of subjeotr. I f a Uneer operator description is t o be attempted, thia Idnear operator expression w i l l Mve va l ld i ty only in discussing behavior with input8 roughly s imilar .to the inputa wed in the measuring experimsnts. Conaequantly, a statistical type of input f'unotion seeme appropriate.

The following l a of neoeasity a vary brief intqodwtion to some of the attempt8 t o dea l with the foregoing problem. Among the ea r l i ea t attem~ptr to charaoterise the human operator's operation were thoee by R. 5. Phi Ups a d A. ~obcayk;' and R. H. Randall, F. A. Rumrell and 5 . R. Raprrini. 4 The ,Phi l l ips equation f o r dbscriblng handuhwl oontrol which uns developed for a study of aided traoking assumed tha t the operator generated a velooity proportional t o the v a l w of the e r ror a d to the r a t e of chbnge of the error a l l delayed by a time lag, L. Thuo the handwheel velocity ,

pH(t) oould be m i t t a n In tm of the eqmr signal Input ~ ( t ) aa followrc

Both b and c a r e parameters, and p is the u o w l di i femnti .1 op ra to r . Rendall a@ his coworkers evolved an expression for dmoribing human operator behavlor which add8 derivative control t o the farogoing caprersiqn so t h a t we have;

The coefficient of the additional t o m ia a p r a m k r , a.

~ w t d s tudied the problan #omowhat more elaboretoly by a r u t i n g a rigM1 oompored of three h a r d a and det.rmining an a p p x i P u t e l inear . law by meens of harmonio analysis of these throe oomponsntr. Trurtln considered the bandwheel diaplaoement and e r m r s i p 1 t o bo oompored of components d s s o r i h b l e by t h i s l inear law a d a rammnt d w to non- l i n e a r i t i w and noiso. TwtIn arrived a t an equrtion of tho fo lhu ing form;

The constant k helped to improve the f i t a t low frequenciee. Using tuo cam apeeda Tuatin studied eight frequencies from .018 to204 c p .

L. ~ u e s e l l ~ in a recent MIT meatarts thesis conducted an e a t e ~ i v e series of experiments in a s t d y of linear approximatione t o human tracking behavior. He used a beet linear model plue noise approach and employed a signal of four harmonica whioh o d d be presented a t three speeds. He thus had a range of from .Q442 to 1.43 c p in the input signal. Russell evolved an eqmtion similar to the P h i l l i p equation. Russell indicated that the husrpn appears to adjust hia response parameters in order t o minimiru, the meen square error in tracking a signal of given power epectrwn. In addition the humen appeera t o s e t hla gain a t a value yielding a reasonable margin of stability. It is well to point out that the parameters found for the four preceding operationel ~prese iona were a l l different. The fac t that a , b, and c differed indicates thet the human's response ia peculiar to given apparatus and display problem. The variation in L from .3 eeconda to .5 seconds is probebly a function of the training of the operators and the predictability of the signal.

North? in a reoent paper on the humen transfer Function, he8 developed a sophisticated framework for the study of human transfer functions. A mean linear operator of simple form is postulated and in addition the cbracter is t ice of a superimpdeed stochastic distribution are discussed. In simple position treickfng North postulates a relation of the form

as the mean steady s ta te operator; and, in addition, he posits a treneient which is kept in a s ta te of continuoue excitation by the stochastic elements. This problem is studied by a epstem of f in i te difference equations as well as by differential equations in v im of the evidence for discontinuity of human operator tracking responses. The discontinuity arises from the fact that there ia a refractory per,iod in tracking since the human appears to prseet reaponsee in accordance with his beet e s t h a t e fro@ available data, and that these responses, once initiated, can not be cksnged continuously. The stochastic process arises from two sources8 one, is inaccurate e a t h a t i d o f the input signal, and two, is the in- ab i l i ty of the operator to perform the appropriate response.

There has been some work done a t . The F'ranklin Ins t i t u t e on the problem of studying human frequency response with the view in mind of obtaining "transfer functions ,I t on a project sponsored by the United States A i r ~orce.6,7,8 The method which th is project has selected fo r studying the human operatorts frequency response is the comparison of the output spec t ra l density t o the spectral density of the visual input using a random signal a s the input. Such a comparison yields utleFul informetion whether or not the human operator behaves in a lineer fashion. Were a l inear aystem under study, it can be ehom that the amplitude part of the systemle frequency response is equal to the square root of the r a t i o of the speotral density of the output t o the spec t ra l density of the input. A complete description of the l inear eyatem would reqnlre the cross spec t ra l density of the output and input so tha t phase as well as amplitude characterietics could be specified. Since the computation of the cross spec t ra l density would have required twlce as much work as the computation of spectral densities, it was decided tha t only the amplitude response would be computed in the preliminary experiments. The main purpoee of the preliminary experimente, the apparatus f o r which was a simple compensatory position tracking device, was t o determine whether the foregoing input-output analysis was a sui table method for analyzing the data of more ambitious experimente using a dynamic f l i g h t simulator f o r a high speed j e t f ighter b u i l t by The Franklin I n s t i t u t e Leboratories. The random input function was obtained by constraining a pip on an oscilloscope t o take positions on a horizontal axis al ternat ively to the l e f t and t o the r igh t of a ve r t i ca l f iduc ia l l i n e so tha t the number of zero crossings were described by a Poisson distribution. In order to make the display somewhat l e s s predictable the amplitudes t o the l e f t and to the r igh t were rtandamly selected from a Gaussian population of the same mean and standard deviation for amplitudes t o the l e f t and t o the right. One of the under- lying reasons f o r select ing a random time ser ies input was the bel ief t ha t the human operator is largely non-linear. This p a r t i c u h r random time ser ies had a specitrum which was similar t o the spectrum of atmoe- pheric turbulence.

In order. t o add psychological i n t e r k t t o the investigation of the applicabi l i ty of the ana ly t ica l procedures, three questions were examined. Does the operator track the random input in a l inear fashion? How does h is behavior change as a function of practice? How do different instructions a f f ec t his response patterns? The question of l i nea r i ty was examined by comparing the amplitude frequency response f o r two subjects when tracking a dis tr ibut ion whose mean absolute amplitude was one centimeter and when tracking a dis tr ibut ion of mean absolute amplitude equal to two centimeters. The effect of practice, i.e., time variation of the system, was exambed by comparing the output of the two subjects when naive and when hi.ghly trained i n following a cer tain random input signal. The effects of instructions to track f o r accuracy and of instructions t o track f o r speed were compared f o r two subjects in order

t o determine if there was behavioral evidence in the output spectra f o r two d i f fe ren t s e t s to respond.

An example of the w e of spectral densi t ies f o r examining behavioral character is t ics can be seen in figures 2 and 3 which i l l u s t r a t e a com- parieon of smoothed spec t ra l densi t ies for'two d i f fe ren t instructions for one highly t rained subject. It rill be noted t h a t I n a t r b t i o n e f o r accuracy resul ted in a considerably smoothed output. The f iduc ia l l i nes represent 2 t confidence bands a r i s ing from instrument errors in the computations of the spectrum.

In general, the experiments were not of su f f i c i en t magnitude f o r an adequate t e s t of the psychological questions raised. On the other hand, the form of the resu l t s indicates the f ru i t fu lness of t h i s method of analysis and its logica l extension t o the interpretat ion of tracking dat'e from the dynamic simulator.

A t present the dynamic simulator has been used f o r a small experiment u t i l i z ing three t rained je t pilots. The data obtained, which is in the form of responses to gust inputs, w i l l be used to obtain cer ta in order of magnitude approximetions t o "humen transfer functions" using The Frenklin In s t i t u t e Advanced Time Scale Analog Computer. The general research program f o r the study of the human operator envisione the use of special data reduction equipment t o be constructed a t The Franklin In s t i t u t e f o r the determination of the pertinent spec t ra l densi t ies and cross spec t ra l densities.

7

135

REE-CES

James, Nichols, P h i l l i p : Theory of Servomechanisms. Radiation Laboratory Series, Vol. 24, 1947, pp. 360-368.

Randall, R. H., Russell, F. A., and Rsgassini, J. R.8 Engineering Aepecte of the Humen Being ae a Servomechaniem. Unpubliehed mimeographed no tee, 1947.

Tuatin, A.: The Nature of the Operator's Reeponse in llanuel Control and i ts Implications fo r Controller Design. Journal Ins t i tu t ion of E lec t r i ca l Engineem, Per t 11 A, No. 2, Niay 1947.

Russell, L. : Characterietics of the Human a8 a Linear Servo-Element. Sc.M. Thesis, Mass. Inst. Tech., Cambridge, Mass., May 1951.

North, J. D.: The Human Transfer Function in Servo Systems. Minis t r y of Supply, Report No. ~iRD/6/50, November 1950.

Krendel, E. S.: A Preliminery Study of the Power Spectrum Approach t o the Analysis of Perceptual Motor Performance. UW1, Vright- Patterson A i r Force Base, Dayton, Ohio, AFTR 6723, Oct. 1951.

Krendel, E. S. : A Spectral Density Study of Tracking Performance, The Effect of Instructions. Uw, Wright-Patterson Air Force Base, Dayton, Ohio, !uADC TR 52-ll Part 1, Jan. 1952.

Krendel, E. S. : A Spectral Density Study of Tracking Performance, The Effects of Input Amplitude and-Practice. U W , Wright- Patterson Air Force Base, Dayton, Ohio, TR 52-11 Part 2, Jan. 1952.

SYNTHESIS OF FLIGHT CONTROL SYSTEMS by DR. JOHN G. TRUXAL

of Purdue University Lafayetfe, Indiana

During t h e past year Purdue University has been privileged t o have a contract from the Office of liaval Research t o invest igate design techniques f o r power boost systems fo r a i r c r a f t control. We a t Purdue f e e l t h a t t he university research groups can not, and should not, be in d i r ec t o r ind i rec t competition with i ndus t r i a l research. Rather we f e e l t h a t t h e primary job of university research should be t he development of basic synthesis techniques.

In l i n e with these aims, our pr ima~g in t e r e s t has been in t h e investigation of t h e basic general problem of t he s tab i l iza t ion and design of power boost qystems. He have attempted t o c lar i fy and del ineate t h e problems involved in this design and t o develop and determine those design technique8 which wi l l be generally applicable in this problem. We have not worried par t icu la r ly about specialized methods of s t ab i l i za t i on which might be applicable t o one a i r c r a f t , not t o another. Rather, our main concern has been with methods which we hope will be generally: applicable; in t h e absence of any one spec i f ic problem, we leave t he more specialized techniques t o t h e i ndus t r i a l research groups.

What i s the fundamental problem? A s we see it, the problem a r i s e s from a basic conf l ic t between speed of response and s t a b i l i t y . On t h e one nand, we have the speed of response of our control system; on t h e other, t h e s t a b i l i t y of t h e overal l a i r c r a f t . The re la t ive location of these two confl ic t ing requirements depends on a number of factors--e.g., the a i r c r a f t speed. A s long as we a re dealing with low-speed a i r c r a f t , there is essent ia l ly no problem. A t low-speeds, a re la t ive ly sluggish a i r c r a f t response i s permis- sible. In addition, t he aerodynanlical character is t ics of t he plane i t s e l f may be re la t ive ly stable. however, as the required a i r c r a f t speed increases, two things s t a r t t o happen: the speed of response must be increased; t h e r e l a t i ve s t a b i l i t y of t he a i r c r a f t decreases. Inevitably, a s we demand b e t t e r and be t t e r perforlance frdm our a i r c r a f t , t h e problems involved in rea l iz ing a system which i s suff ic ient ly s tab le and a t t he same time responds with adequate spee4 becomes more d i f f i cu l t . 'tie f e e l tha t we a r e now a t t h e point where design has become d i f f i c u l t , p a r t i c u l a r u i n those cases i n which we dea l with the a i r c r a f t a s a gun platform. Essentially, the s i tua t ion here i s no different t t u t ha t i n ~nany f i e l d s of engineering. T P i s i s a universal phenoniena--this development of t au t e r and t au t e r specifications, a s t h e years go by; t h i s demand, a s these specifications become more tau t , f o r a more complete understanding of t he basic system--and in p a r t i c u k r a f u l l e r understanding of what fac tors influence the system performance.

Chw f i r s t concern then has been with a de ten~unat ion of those fac tors which do determine the s t a b i l i t y and the speed of response of the a i r c r a f t s

with a control system actuated by a poner boost system. The syatsan we have considered i s shown in Pig. 1. Our cgr6tam consi8ts basicaUy of five parts. Working from the stick t o the control surface, from lef't t o right on the diagram, we have first a mechanical coupling system, between the s t i ck and the hydraulic power qystem. k here represents the compliance of the push rod. or control cable from the s t i ck t o the input t o the Wdraulic gyetm. The damper f p aad lnrrsr % represent the impedance seen looking into the input t o the-bdraulic Getem from the output d of the control cable. In our initial &udy of the qystm, ws have a 8 d that these parameters are irdqandent of the gain of the Wdraulic system. I will have &re t o say ooneerning these oimplifying arsumptionr later .

The inprt t o the hydraulic system i s the displacsmsnt xp. We consider a singlbstage m a u l i c system, with an output displacement + actuating t h e be l l c r n k t o give a deflaation of the control surface & . The normal, or vert ical aeceletration of the airplane, n, i s detarmined by this deflection through the a e r m c s l characteristics of the aircraft. The a r t i f i c i a l f a e l ~8 'km Vb is the 8-108t po88ibh8 C O M ~ S ~ * OW of the bobweight of mas8 rmnll n. We do nat Imply that t h i s is a t a l l typical of general a r t i f i c i a l f ee l ayrtaw. Certainly our system probably should include a t least a spring. Howwer, we fee l very strongly, as I shall txy t o point out la ter , that the d e s i s of a wi table a r t i f i c i a l f ee l qpstsm depends on consIdaration8 of dynamic &ability (and when I speak of stabil i ty, I refer t o d p a d c stabil i ty) and the nature and characteristics of the pilot in the closed loop system. We feel , and perhaps you will agree as wd continue, that this simple bobweight system suffices as an emample of the general prooeduree and ideas I hope t o present t o you. The introduction of a spring does not, for e ~ m p l e , materially change our development.

Figures 2, 3, and 4 show the basic components of our syetem in more detail. Figure 2 shows the aerodynamic axes a& basic equations, Plg. 3 the stick dynamics and Fig. 4 the hydraulic qystem. The pertinent equations accontparUr the figures. A considerable portion of the &fort of the Purdue group over the past year haa been spent i n the aualyds of the bdraul ic system, the development of m e a s u r i n g techniques and the correlation of valve configuration with s ta t ic and dynamic characteristics. Homera today I would l ike t o consider the hydraulic valve as described simply by the flav-proportional- to-displacement equation, which is an adequate deecription over the frequency range of interest t o usa and oadt any detailed disoussion of other possible linear and nonlhear analytical descriptions of the system.

Referring again t o figure 1, ue see that there are basically two feedback eystcms t o be designed-the &draulic system, a c h is a closed-loop, posi- t ional feedback ggstem, and the overall airplane control system which contains the 4ydrauU.c loop ao one co~lponent. OPT i n i t i a l interest a t Purdue was concentrated on the w a u l i c system, but it rapidly became apparent that this one component could not be intelligently designed or andlysed without c~neideration of the overall qyatem. I mid l ike to speak briefly about the stabilization of the m a u l i c system and then consider in more deta i l the desi- of the overall loop.

The hydraulic system i t s e l f is a high-gain feedback ayetern. The load parameters of t h e hydraulic qystem a r e the effect ive mass of' the elevators, t h e aerodynamic spring r a t e and t h e damping, both t h e aero- meal damping and tha t associated with the =in power piston, In addition, the compressibility of the o i l ac t s essent ial lg in shunt with the main load, since the flow of o i l can go e i ther i n to compression of t he o i l or i n to motion of the main piston and control surface. Up t o reasonably high frequencies, typical ly in the order of magnitude of 15 cps o r above, t h e open-loop hydraulic system integrates, or, if we look at it from t h e frequencp-response standpoint, the gain drops off inversely proportional t o frequancy, Aa we reach the frequency a t which resonance oocure between t he o i l compressibiUty and the load, t h e output tando t o inorease. In other words, in t h i s b a of frequencies t h e aofnpre5aibUty ef fea t i s such that it tends t o increase the motion of t he main piaton. When the main piston is moving t o the right, f o r example, t he o i l in t h e l e f t cylinder i s expanding, tending t o give t h e piston motion an extra push. This resonance phenomena is very l i gh t ly damped due t o the low f r i c t i o n a l losses i n the systam. When the hydraulic loop is closed, through the input -age, the system w i l l ordinarily o s c U t e with an amplitude driving t h e system i n t o t h e nonlinear region of oscillation. In certain cases, these nonlinearities, inherently preaeat in the system, or t h e f r i c t iona l losses which we have neglected tend t o s t ab i l i ze the wetem, but the re la t ive s t a b i l i t y is poor.

Thie resonance phenomena, and the consequent ins tab i l i ty o r poor r e l a t ive s t ab i l i t y occurs a t such a hi* frequency compared t o t h e siepi- f icant frequencies of t he remainder of the overal l aerodynamical systcrm, it doe6 not appear troublesome on the surface. The main d i f f i c u l t i e s a r i s e because these frequencies are , however, of the same order of magnitude as t h e f l u t t e r frequencies of t he a i r c ra f t . The coupling between the f l u t t e r system and the Wdraulic and control system may cause undesirable amgUfi- cation of the f l u t t e r amplitudes.

This coupling between the unstable hydraulic system and the f l a t t e r system and the resul t ing tendency t o increase f l u t t e r troubles can be c i r c u w ~ t e d in e i the r of two ways. First, we might simply cut off t h e Wdraulic wstem a t a frequency mch lower than the band of frequencies of significance 'in f l u t t e r phenomena. Seconily, w mi@t t r y t o increase the re la t ive s t ab i l i t y of the hydraulic system while maintaining t h e high low- frequency gain. The first solution is certainly the simpler. The principal disadvantage of this method of cutting off t he wdraul ic system a t low frequencies is the resul tant sluggiah response of the overal l aircraft system, o r even of the wstem from the s t ick t o t h e control cnrrface. This relationship betwema the bandwidth and the tims d e 4 o r time of t h e t ransient response is nothing quantitative, but in general it can be said tha t , i f t h e aystem i s suitably damped so tha t there is not undesirable overshoot in the t ransient response, the time required fo r the wstem t o respond t o a step function input and essent ial ly reach the steady s t a t e is inversely p r o p o r t i d t o the bandwidth. In part icular , if t he bandwidth is one cps, a s an example, the system will respond t o a t ransient input in something like one-half t o one

second, a t l ea s t t o the proper order of magnitude. Thus, if we attempt t o r id ourselves of the trouble with t h e re la t ive or absolute in s t ab i l i t y of the hydraulic system by cutting down the bandwidth, we soon reach a point where the time of response of t h i s part of the overal l system becomes undesirably long. I n one sense, t h i s i s indeed an unfortunate circumstance, since t h e bandwidth of t h i s part of t he system can simply be reduced by cut t ing down the gain, t he bandwidth being r0ughJ.y proportional t o the gain over the range of frequencies of in te res t t o us.

The second al ternat ive fo r eliminating t h e undesirable effects of coupling between two h i g h l ~ underdamged system--the f l u t t e r system and the hydraulic system--involves stabildnation of t he h y d r a d o system wNle maintaining the hi& gain essent ia l f o r f a s t response. The diagram of f igure 5 indicates the d i f f i cu l t i e s involved here. A s it now stands, t he hydraulic system i s charact e r i s t i ed by a transfer function-the Laplace transform of the r a t i o of the output over the input-with three poles- one a t the origin, and the other two a conjugate complex pair very close t o the jw-axis. If we use the root-locus method of analysis we can see a t once the e f fec t of closing the loop with varying amounts of gain. In part icular , we study what happens t o the poles of the closed-loop t ransfer function of the hydraulic system a s the gain i s increased. For very low values of gain, t he poles of the closed-loop system function a r e ident ica l with those of t h e open-loop t ransfer function. A s t he gain i s increased, t he poles of the closed-loop system function move toward inf in i ty , t h e location of a l l the zeros of t h e open-loop function. The par t icu lar paths taken by these poles a s the gain i s varied a r e shown i n Fig. 6: one moves out from the ori- along t h e negative-real axis, the other two move away from t h e conjugate co rnex pair of poles over in to t h e right-half plane. For a re la t ive ly low value of gain, these poles have crossed i n t o t h e right- half plane, indicating tha t the closed-loop system is unstable.

How CM such a system be s tabi l lzed? There a r e a variety of nays, im terms of t h e pole-zero loc i , and then a number of ways in which these desired changes may be instrumented. We consider only one general method in any deta i l . The furdamental d i f f icu l ty i s the proximity of t h e conjugate poles t o t h e j W d s . This a r i s e s because of t he very sharp resonance betwem t h e compressibility of the o f l and the mechanical and aerodynamical load. This resonance can be msde l e s s sharp, and a l so incidentally moved t o a lower frequency, if the compressibility of t h e o i l can be effectively isolated from t h e load--in other words, if a t t h e same time the expanding o i l i s giving the pis ton an extra push, there exists an al ternate network which is absorbing t h i s extra effect ive flow. One possible system for doing this is shown in fig. 7.

Hare i n Fig. 7 both the hydrauuc-mechanical system and t h e equivalent ' e lec t r i ca l network are shown. In the equivalent c i rcu i t , t he load corresponds t o the ser ies resonant circui t made up of LC, Rc, and Cc, repressnting respectively the mass, damping, and compUance of t h e aerodynamical and mechanical load. The o i l compressibility is represented by the parall- condenser C. The resonance phenomena, speaking i n e l e c t r i c a l terms, i e

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By way of introduction t o this problm, I wauld l i k e t o revert fo r a momsnt t o some of the ideas expressed in the axcellent p a p r we heard on Tuesdey rnornbg by Mr. Andrm of Hughes Aircraft and presmted bgT Mr. Turner. In this paper it was rather forcibly pointed out that we are rapidly rea- the point where we must design our p o r n boo& control system t o meet quantitative speclfioatiom concerning both s t ab i l i ty and control. I l i k e t o think of the situation in something like the foIloving light, with a very s-6 analogp. I picture the w

control system designer in a position somewhat ana log~ls t o a &ern- y e w old boy in a baseball game. An the game start@, the other bags are a l l under ten years old; the mere presepce of our slxtem-year old is emougb t o throw fear into the opposition, Just es in the early years of p m r boost

. system, the mere presence of the control wet= desiepar and a l i t t l e imaginetion on his part solved the problems. A s the game plogresses, however, older boys arrive. Our aixtegt-year-old has t o work t o get his base hits. He begha t o look amuad fo r a better bat and tools; he s t a r t s t o consider using a l i t t l e s t r a t e m instead of the previous brute-fome tactics. In an analogous manner, the control systems engineer starts t o adopt same of the servomechanism theorg--the use of Nyquist diagrams that have been mentioned several times here during the past few days, f o r axample. BLlt unfortunately, the game has not yet s t a b W e d . For, approaching our ballfield are a half dozm professioaal b a l l players, eager t o get into the game. Our s M e m - year old is wing t o have t o call on a l l his abi l i ty i f he is t o hold his own. In our inmediate problm, we see these professional b a l l players a s the problems associated with our planes f lying a t a Mach number of &5. If we are t o design successflrl power-boost aystams and tactically-capable a i rcraf t operating a t these high speeds, it seerme imperative td us a t Purdue tha t all available knowledge be brought t o bear on the problem. We f e e l tha t in a few years, when we are confronted with the design of the power boost systems for these high-speed aircrafb, there will be no accuse for us not h a m a understanding of the problem and a general idea of a suitable approach t o the aolut ions.

What specifically are the problems associated with the design of a high- performance power-boost wstem? Thw are - in number. We wish t o mention only a few of them today in any detai l . The problem we have heard most about a t this meeting i s the design of an appropriate a r t i f i c i a l f e e l system. I feel, howemr, tinat this i s only one of the basic problems. It has received considerable at tention here during the past few days principally +cause it i s a problem which is already confronting the engineer. Of a perhaps !laore c

basic nature, however, are the twin problem of s tab i l i ty and control. Indeed, the satisfactory solution of the a r t i f i c i a l f ee l problm can only be arrived a t in conjunction with a t leas t pa r t i a l solutions t o the stability and control problems, for the a r t i f i c i a l f e e l system is an integral part of the overall

- loop which does determine both the relat ive s tabi l i ty and the control charact'er- i s t i c s .

What essentially are the problems associated with control and s tabi l i ty? The most pressing problem, it seams t o me, is the development of a more

exp l i c i t set of specifications-or a more quantitative description of the behavior we desire from our system. Some of you ~lvyr say that quantitative specificstiona can not be written--instead we nust be sat isf ied with euch specif icat ions a s the statement-The airplane mst fJy sat isfactori ly . This f i e l d of engineering i s not alone i n t h i s d i f f icu l ty of writing a cpanti ta t ive se t of specifications. The seruomechaniem f i e l d in general i s plagued by the same d i f f icu l t ies . I have heard maqy tin108 t h e et,atemsnt-- We want a real f a s t servo tha t has pract ical ly no aror--or somuthing equally vague. If there is t o be logic t o the eyetom design, t h e f i r s t s tep involves making this specification quantitative. Indeed, when this is done, the problem is half solved.

In particular, what specifications a r e appropriate t o the aircrai"t control problem? We consider still, f o r t he sake of explicitness, only the longitudinal control system. 1i-1 addition, we consider o n . the performance specifications, taking the word performance i n i t s conmon, narrow meaning, and excluding such factors a s the maxhm~ allowable acceleration, etc. We consider only specifications governing t h e e t ab i l l t y and dynamic performance i n general of the a i rc raf t undsr small daflections of t h e control w f a c e .

The specifications, it seems t o me, should include a t least the following quantities: t h e re la t ive damping of t he overal l a i r c ra f t system, t h e damped (or undam~al) na tura l frequencies af osci l la t ion, t he low-frequency gaia of t h e system, and t h e t h m lags i n the various parts of t he system. The firat two quantities, r e l a t ive damping and osc i l la t ion frequency, dot elmine t h e re la t ive stability . The t h i r d quantity, t h e low-f requency gain, datemines, essentially, t he effectiveness of the feedback, the ab i l l t y of t he systan t o continuously ca l ibra te i t s e l f , reduce the effect of nonlinearities, aDd control t h e effect of corrupting s ignals entering t h e system ( fo r example, wind gusts). Thg last quantity, t he time l ag In t he various system components, measures t h e speed of response of our a l rc raf t .

Other quanti t ies will of course have t o be specified qventuaUy,in t h e complete problem. For example, we a re interested not only In t h e low-frequency responee of t he system t o wind gusts, but a l so the overshoot and s a t t l h g timb associated with t h e response t o a gust of wind. However, t he above fou r parameters will give us a basis f o r t h e remainier of this paper.

Ql t h e b a d e of specifications of t h i s sort , at l eas t t he prelhhaxy design of the syetm can be aarried out on a logica l basis. The dismssion here is samewhat d i f f i cu l t , because the two aepecte of control and s t a b i l i t y can not be divorced, but I sha l l t r y t o indicate t h e general t r a of a method which we think is reasonable f o r design. Mrs t , l e t us consider the s t a b i l i t y problem. does the system basically tend toward in s t ab i l i t y? And does t h e s t a b i l i t y problem become more d i f f i cu l t as t h e speed of t h e a i r c r a f t increases. In the very simple system we a r e conaideriag here (shown

' in Fig. 1) signif icant phase lags In the system a r e introduced from two

reutione, the aerodynamics and the contrdl stiok dynamicre If the phugoid motion and very low frequency behavior i s neglected, the a-cr behaver essmtially a s a sewnd-order rgrstem, with a tranefer f'unction of the form ahom in SQ. 1. The relative damping rat io assoaicrted wlth the conJugete pole8 here i r in our arample about 0.4. At'high apwds, this relative damping ra t io could be axpected t o go or low as 0.1 or lesr. Additional phaae lag is introduced i n the uontrol st ick dynamics, the medwiaal vtan f r m the control otick t o the input t o the hydraulic ~grrt-. The ~grutm wr ere oonaidaring here i r ahom in Fig. 3 atxi ha8 en rppmxbate tranafer funotion af the form: -

The overall open-loop systsm th& has a transfer function proportional t o the product of these two individual functions.

K Sfs+8.Z)(sZ+ 7:l.s tC3)

The corresponding pole configuration for the open-loop wstem i s of the f o m shown in P i g o 10.

We now consider what happens as we close the loop and bring up the loop gain from a very low value. The poles of the closed-loop transfer function move in the maaner shown i n Fig. U. For relatively low values of loop gain, the closed-loop system is unstable. How can t h i s ~syotem be stabillzed? In particular, haw can the system be stabilized a t a value of loop gain vrNch is mfficie~lrtly high to insure that we have adequate control characteristics? The first and mgot obvious method is t o introduce a device t o cut the qotem off, a t least part way, a t very l o w frequenciee-a device which the electronic engineer wuuld term integral control. Such a device necessarily tends, as we saw earlier, huuever, t o decrease the speed of response of the system. We have f e l t that the primary Job of any stabilization that is introduced should be t o force the airplane t o respond more rapidly than the aerodynamical uoefficients would ordinarily allow. Certainly, a t least, thq stabilization system should not make the system more sluggish.

Considering this pole configuration, one possibility is irmnediately evident. We might attempt t o move the pole due t o the stick dynamics, wr at-8.2, so f a r out into the l&-half plane that the system essentially reduces t o a third-order r i ther than a f ourth-order system. T h i s can be readily accomplished a s shoun in Fig. U simply by an increase in the value of the damping.coeffic5mt fp. There are two difficult ies that arise a t once however, with such a scheme of part ial stabilization. Flrst, the force required of the pilot to move the stick a t a constant velocity, when he i s

- moving essentially against th i s damper, increases intolerably. Secondly, the time lag betwleen the stick and the output of the hydraulic ~ s t e ~ increases proportional t o th is damping coefficient. In particular, this time lag, assuming the hydraulic system acts instantaneously, is given tgr the rat io of the damping coefficient, fp, t o the spring constant of the push-rod, k, a s is shown in appendix A. Thus, about the beet we can do

her@ is t o lnake the push-rod a s s t i f f a s possible, and increase the damping f p as much a s pemissible and then l ive with these components.

It appears then that in the absence of the abil l ty t o effect arg 'remarkable isprovaments i n system components, the stabilloation of our system with the simultaneous realization of high performance characteristics 4

demands the introduction of additjanal components. We ask ourselves what distortion of the pole confiwation we mw have for the opm-loop system would result in desirable characteristics. One part icuhrly si@e configuration is shown in Fig. 13. We leave the two poles due t o the - stick dynamics a t the origin ard -8.2, but move the conjugate compleeg poles due t o the aerodynamics either onto the negative-real ax i s or a t least well out into the left-half plane. Now what happens as we close the loop and increase the gain? The poles way out t o the l e f t are of l i t t l e interest t o us. The imporLant loci are those stemming from the two poles on the negative-real axis and close t o the origin. A t low frequencies, these poles move as a function of gain in the manner shown in Fig. U.

the addition of a zero a t -10 and the moving of a l l other poles way out into the left-half plane we have kidded these two poles into thinking we have a configuration of the farm shown i n Fig. 15. It is not unt i l we reach reasonably large values of loop gain and fa i r ly high frequencies, that these two poles realize the deception and bend over toward the right- half plane. Then i f we keep the gain a t a value corresponding to these poles located a t the points A and 81, we have a closed-loop system, an overall aerodynamical system, uhich behaves eesentially as though it had t h i s pole-zero configuration. The response i s characteristized by one pair of conjugate complex poles a t a reasonable damping ra t io and a dipole on the negative real axis. The transient response will be roughly of the shorn in Fig. 16.

dseenthlly the earns results can be achieved by a slightly different, and basically more logical approach. Suppose we simply start off by saying that we want a transient response of the form shown in Fig. 16. Int us write the overall sya tm function, the Saplace transform of t h i s transient response. We denote t h i s desired closed-loop transfer fundion as ~ ( 8 ) . horn this ~ ( a ) we determine the required open-loop transfer function, which we migbt ca l l p/q, the rat io of two polynomials in s. T h i s opan-loop tranafer function, ~ / q , i s related t o ~ ( 8 ) a8 s hom a t the bottom of PZig. 17. . p and q, the ~xumrator and denominator polynomials of the open-loop tranafer function are readily determined from t h i s relationship. p is simply the numerator of ~ ( 8 ) . q i s simply the denominator of ~ ( 8 ) minus the numarator, and can be determined in factored form by a simple graphical subtractian, - plotting the polpomials for negative-real values of the variable s. A plot of this form is &awn above the equations in Fig. 17.

Let us review for a moment., We have said that our fundamental philosophy i n the design of the power boost system' should be something l ike the following: We f i r s t decide on suitable specifications for our

91 '))M o=oa WTM F'

om* 03 eeuodra me*&

particular problem. We then choose a transfer f'unction which yields these specified characteristics. The last step in the design involves the instrumentation of a sgstau which will yield our desired transfer function. Several of you w i l l undoubtedly object t o this general approach which re are postulating. me principal objection possibly wlll be that we can not in general force our aerodynamical system t o behave in any we want. Wa know, for exi%&ple8 that we can not d d an overall widean function of unity in practise. That is, we can not expect to realise a power-boost control stick aption, with absolutely no time lee. However, we are obviously being ridiculous in asking for performance of this qua,lity. Clearly, m, are U t e d i n attainable performance by the saturation of the control surface effectiveness. What our stabilisation and' e q u a m t i a a sgstau is doing i s anticipating the lago axi instability inherent in the a e r o d y d o s and stick dynamics and attempting t o compensate.

Up t o this point the design of the cprotsm is straightforuzmi, follawing l ines w e l l established in the servomechanism field. (I of course do not mean t o imply simplicity by the mrd straightfommi). If the control system is e l e c t r i a l in nature, or a t least contains certain electronic wmponents, the compensation can be conveniently and readily introduced. However, i f the control system i s mschanical-hydraulic sy s t a~ , the design of suitable compensation networks is not as straightforward. The Purdue University group is now investigating a &or of lnechanical and hydraulic circuits which look promising for the accomplishment of t h i s compensation. The design of comgcmsat$oa networks is made diffioult by. several factors which can not readily be expressed enalytioallg, but which are of paramount importance i f the networks are t o be of practical value. First, we hew f e l t that comparnsation mst be accamplished simply. We do not fee l that rrr, can introduce, for exa@e, a large number of nonlinear circuits or complicated circuits. Secoadly, the synthesir taf suitable mschanical-k@raullc networks is complicated by the requirements on the allowable forces against which the pilot will have t o work. In other rrords, networks introduced in that wction of the system which we kve called the stick dynamics ~ilrst not lead t o rmdesirable forces referred back t o the d i c k bandle. Thirdly, it is verg difficult , I f not' hpOSSibl@8 t o &idpa te d i i f i c ~ l t i e s With distributed fri&i0n8 &C b the last aaalysis, the propriety of any system mrst doped upon W a . 1 tests. A t this t-, we have not yet reached the point where we are able t o present experimental data.

- The basic type of compomtion which seema t o be most d f e c t i r s theorat- icalls is a lead form of network. A generalized. form of tranafer function f o r th. desbed conrpensation network ;auld of the fomt

sa+2&d,,a +w, 4 4 b s t t a s +A*

Ye have spont eorm time investigating ,the poesibility of instnmnanrting mechanically and hydraulically a transfer function of this type and f ee l that a twin-T qgetga of the fora shown an Fig. l8 holds eoneiderable prauise. The mechanid sohematia ia shown at the top of the page and a & d o h of the q g s t a ~ at the battm. Referring t o the sketch, the top half of the twin-T is realiaed by the center tm springs atxi the orifice, the battau U of the twin-T by the tm end orifices and springs and the hydraulic

coupling between the tm ends and through the l i n e shown above the main par t of t he figure. The output spring, shown in. the schematic is not shown on the f igure a t the bottom of t h e page.

There are a number of other systems which accomplish essent ial ly the same resul ts , but the d i f f i cu l ty of real izing a decent range of element values has led us t o consider a schane of this sor t in s o m deta i l . For * t h e s tab i l iza t ion of the control system with the numerical parameter values used a s an example during t h i s paper, t h e spring and damper values f o r t h e twin-T shown here coma out t o be of order of magnitude yielding time constants of about 1/10 secord. Specific values d e p d on t h e location in the ~ystem. The element values can be readily determined'using the methods of network synthesis well known i n e l ec t r i ca l engineering.

Clearly a findamental question is the proper location of thls twin-T o r any other form of compensation network in the overal l loop. The answer t o thls wes t ion depends on a number of theoret ical fac tors (e.g., the effect of loading a t e i ther end of t he network on i t s characteristics, t he effect of the loading on t h e r e s t of the gystem produced by the network, etc.) a s well as such prac t ica l fac tors a s space avadlabili ty, e f fec t of fai lure, etc. One possible scheme is shown in Figure 19, where the compensation network i s inserted a t the input t o the hydraulic system.

Two comments a r e appropriate a t this time concerning the stabilization and compensation concepts discussed in t he past few minutes. Eirst, it i s clear now, I hope, why we f e e l t ha t t he en t i re wes t ion of a r t i f i c i a l f e e l is one which can not be divorced from the s t ab i l i t y and control questions. In the airplanes which have been discussed during the past few days, t he performance character is t ics have been suff icient ly low, i n mimy cases, so t h a t t h e a r t i f i c i a l f e e l system presented a more or l e s s isolated problem. However, a s the sat isfact ion of performance specifications becomes more d i f f i cu l t (and we do not envision t h i s a s being too far in t he future) t h e a r t i f i c i a l f e e l system and the character is t ics of the p i l o t (or a t l ea s t some generalized description of these character is t ics) must be of basic importance in determining system s t ab i l i t y and control labi l i ty .

If the s t ab i l i t y and control labi l i ty problams a r e t o be resolved without unnecessary d i f f i cu l t i e s , the f e e l system must be designed a s part of the overal l loop.

The second comment I would l i k e t o make a t ' th i s time concern8 t h e e f fec t of nanl ineari t ies in the various system components on the s tab i l iza t ion procedures we have discussed. ~ h e s e nonlinearities are, a s I see it, of two general types--the nonlinearities of specific hardware, a s the hydraullo system. Xe f e e l tha t nonlinearities of t h i s type can be controlled and kept small enough t o make a l inear analysis and -thesis a t l ea s t a good first order approximation. In the l a s t analysis', f inal adjust~nent must of couree be done axperbentally. The second type of nonlinearity i s perhaps more troublesome. I refer t o the changing aerodfnamical coefficients Kith

changing f l i g h t corditions. Idea-, one would design dynamic compensation networks, a s suggested in some of the e a r l i e r talks--compensation networks which would be automatically compensated f o r c-ng longitudinal velocity, e ta . If the compensation i s effected ent irely electr icalJy, t h l e dynamic cornpensat ion can be approximated fairly well, although t h e complexity of t h e instrumentation necessarily i s greater than with a l inear qystem. If as we have t r i e d t o suggest in the past f m moments, t he campensation is effected hydraulically and mechanically, dynamic compensation is a more d i f f i c u l t problem. Ae y e t we have done nothing along this l ine.

I would l i k e t o summarize briefly. We f e e l t h a t t h e time is rapidly approaching when the s t a b i l i t y and control labi l i ty problems of design w i l l be of paramount importance. We f e e l tha t it is essent ia l tha t logica l design procedures be established in anticipation of these difficulties.

log ica l desiep p r o d w e must proceed from quantitative 8pecifications. It must lead t o sui table general networks f o r accomplishing t h e cornpeneation. We f e e l t ha t conpensation system can be determined which will essent ial ly force the airplane t o meet t he selected performance specifications. The rea l iza t ion of these compensation systems can be accoqlished electronically. We al8o f e e l that compensation can be effected hydraulically and mechanically. We believe tha t there i s a great deal of work yet t o be done i n evaluating various Q-draulic-mechanical systems f o r compensation, in studying t h e e f fec ts of various locations of these networks in the overal l system, in considering t h e poabibill ty of using pa ra l l e l compensation, and in the related probldl~lb.

The system. I have ueed throughout t h i s paper a s an ~ l e of some of these ideas is about a s simple a system as one can devime t o scoomplish even theoret ical ly t h e desired control. We f e e l t ha t it is essent ia l t h a t mare complar (and r e a l l a t i e ) system8 be considered from this standpoint of s tab i l laa t ion devices. appreciable complication of t h e system w i l l necessi tate going t o computer and simulator studies, a s the introduction of additional degrees of freedom rapidly places the complex it^ of t he wstsa beyond t h e bounds of human comprehension. Thus far, we have not done Whlq appreciable with d o g computer studies, a s we have f e l t t h a t in te l l igent us. of ccenputer data required a sol id understanding and background in t h e behavior of simpler elystams.

Int e ~ p r e t a t ion of Specificat ions in Tenas of Syat em Parametera

Specification on low-2requency loop gain:

A t low frequencies, open-loop transfer function behaves a s $ i s direct ly related t o the polee and zeros of the closed-loor,

t ransfer function:

Specification on time lag between s t ick mt ion and nrotion of control -face:

Denote transfer function from s t ick motion t o control surface as:

= J( s ) x s b )

Then with constant velocity input, time lag (steady-state) is:

with our unstabilized system

specification on s t ab i l i ty r

A specification that the short period dynamic osci l lat ion of normil acceleration shall be damped to 0.1 amplitude in one cycl* msane t ha t if the response i s primarily controlled by one pair of condugate complex polee, the associated re la t ive damping r a t i o should be greater than about 0.37.

4 i) .So Air Force Specification No. 1815-B

A,. Area of bydraullc jack giston. c, ceJ cn# &S %

Parameters In electrical system nrrrlngcrus t o bydmullc aystsp! vith compensation.

f~

Damping coefficient of mechanical linkages.

f n ~ ~ m p i n g coefficient of b y d r a u ~ c agrstcm canpamator.

c Damping factor i n hydraulic systsla loed.

pa Bob weight acceleration force. 8 Aoceleratiata of gravity. H ( 8 ) Overall system finetion h(s) Denanbator polynomial for H.(B) J(s) *stem function from stick t o control 8uriaee. K Gain constant. k Spr3.ng constant of prsh rod from stiok to qydrrulla qpd& kc A-c force spring rate. k e OF1 oompressibility spripg rate.

KQ Qnirau3.i~ e l y a t a n control valve gain.

k, Spring constent of wraallc qgstam wmpn88tioa nstworlr.

=, Velocity constant. 1 Born we-t mss. Wp Msc@anicsl linkage mass.

n, Cantrdl column mass.

Hc EXfectiw mass of elwator.

n, k e s of 4pdraul.i~ system ampmeation nstmrk. n &ad factor. P Pressure across hydraulic jack. PI, p2,. . Negative of poles of t r anda r hnctions.

~ [ s ) Nunrrtrator po- far ~ ( s ) . s 6) ~ e m d n a t o r polyr~~ndal fool opgl loop t randor hmcticm. fa N e t ra te of flow through bdraul ic qpstm aatrol valm. 8 Isplace transform variable t Timb

Tlag Time Lag from s t ick t o cc4rtirol surface.

U Average f OM airplene velocity.

5 Inpllt t o hydsaulic Iaystmn

=c Output displacement of m a u l i c qstm.

% Valve displaament Control coluaP diaplauenmt

]b Mechanical linkage displaaaernnt.

*buenbeag quwuooea pmqm pedmpun *s *Lbaenbeag xe~dmao go q~8d &wu@mq 'b

(lp *s #Jbuenbeg xe~&oa go w8d -8 9

'qaq~d JO 8 *oT'?wJ 9-~BP 9hTqQTeE

*.xoqme JO uof qom J~IVUV t 'W8qq8 30 e@I8 UOfq8qJllvbd "P *

qgzyen qoq JO quameoqdsy(l 2 *uo~qoury Jejsmaq JO some 30 e~~qw9e~** 'Za #Tz

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